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The elementary divisors of incidence matrices between certain subspaces of a finite symplectic space
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Lataille, Jeffrey Michael, 1973-
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Thesis (Ph. D.)--University of Florida, 2001.
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Includes bibliographical references (leaves 63-65).
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by Jeffrey Michael Lataille.

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Full Text

THE ELEMENTARY DIVISORS OF INCIDENCE MATRICES
BETWEEN CERTAIN SUBSPACES OF A FINITE SYMPLECTIC SPACE

By

JEFF{EYi21-ACIIAEL

LATAILLE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2001

ACKNOWLEDGEMENTS

I would like to thank my advisor Dr. Peter Sin, to whom I am grateful for his help and guidance. I would also like to thank Dr. Chat Ho, Dr. Pham Huu Tiep, Dr. Helmut Voelklein, and Dr. Meera Sitharam for attending several of my talks and for offering helpful comments and criticism.

ACKNOWLEDGEMENTS ...............

ABSTRACT .......................

CHAPTERS

1 INTRODUCTION ................

Description of the Main Problem ..... Statement of the Main Result ....... Background on Incidence Problems .... Outline of the Solution ............

2 REPRESENTATIONS OF G IN ODD CROSS CHARACTERISTIC .

2.1 Preliminaries ..... .............................
2.2 The Complete Submodule Lattice .................
2.3 The D im ensions . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 REPRESENTATIONS OF C IN CHARACTERISTIC 2 .........

3.1 Preliminaries .....................
3.2 Restricted Action ...... . .....
3.3 The Composition Factors of RessGM k02 . . .
3.4 The kGM-Composition Factors of k2 ....
3.5 The kG-Composition Factors of kcl. .......
3.6 The kG-Submodule Lattice of k" ........

4 DETERMINATION OF THE p'-TORSION ....

4.1 Preliminaries .....................
4.2 Determination of the p/-torsion in coker 7r,1 .

5 DETERMINATION OF THE p-TORSION ....

Preliminaries ..................................
Related Structure Theorems ........................
The Modules M................................
Determination of the p-torsion in coker q,- ..............

REFERENCES ...........................

BIOGRAPHICAL SKETCH ............................ 66

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE ELEMENTARY DIVISORS OF INCIDENCE MATRICES BETWEEN CERTAIN SUBSPACES OF A FINITE SYMPLECTIC SPACE By

Jeffrey Michael Lataille

December 2001

Chairman: Dr. Peter Sin
Major Department: Mathematics

Let p be an odd prime and m > 2 an integer. Let V be a 2rn-dimensional Fp-vector space equipped with a non-singular alternating bilinear form ( , ). For 1 < r

CHAPTER 1
INTRODUCTION

1.1 Description of the Main Problem

Let p be an odd prime and let V be a 2m-dimensional Fp-vector space equipped with a non-singular alternating bilinear form ( , ). We say that the form induces a symplectic geometry on V and we refer to V together with (,) as a symplectic space. If rn = 1 then V is called an hyperbolic plane. We exclude this case from consideration and shall always assume that m > 2.

We say that the vectors r, i, c V are orthogonial if (r. u,) = 0. Given a subspace R C V we define

R':={vEV I (v,w) = 0 for all w E R}, and call R' the orthogonal complement of R. A subspace R C V is called isotropic if R C R'. It is a standard fact from linear geometry (e.g., see Artin [1]) that V has an isotropic subspace of dimension r if and only if 0 < r < m.

For 0 < r < n, let Cr denote the set of r-dimensional isotropic subspaces of V and let C2m-r denote the set of orthogonal complements of r-dimensional isotropic subspaces of V. There is a natural incidence relation between any two of the sets ï¿½Cr and C, given by inclusion of subspaces. We can encode this information in an incidence matrix as follows: Fix integers r. o such that 0 < .s < r < 2m. and assume that we have some fixed but arbitrary orderings on the elements of 4C. and 4, so that we may write
a = {RI,...,RiCr}

and

Define a ,CI x 1,,l matrix r =r,s (aij) by

ai~j f if Si C Rj"
{0 if Si Z R,

We call r,s an incidence matrix between ï¿½r and L,. It is easy to see that if 71, is another incidence matrix between 4 and 4, (i.e., with respect to some other ordering on the sets 4,. and 4,), then r, = P,.,,Q, where P and Q are permutation matrices of appropriate sizes. Since all of the entries in 77r,s are either 0 or 1, we may regard rlr,s as a matrix over any commutative ring. We will be interested in regarding Tr,s as an integer matrix.

We pause now to recall some basic facts from matrix theory (e.g., see Newmann [33]). Two integral matrices A and B (of the same size) are called equivalent if there exist invertible integral matrices C and D such that B = CAD. Each integral matrix A is equivalent to a matrix of the form diag(dl,... , dr, 0,. . , 0), called the Smith Normal Form (SNF) of A. Here each di is a non-zero integer, di divides di+l for all 1 < i < r - 1, and r is the rank of A. The elements {dj} are unique up to associates and are called the invariant factors of A. The prime-powered factors of the invariant factors, counted according to their multiplicities, are called the elementary divisors of A. A knowledge of the invariant factors is equivalent to a knowledge of the elementary divisors.

Regarding 77r,, as an integral matrix, it is natural to ask for its elementary divisors. We remarked above that all the incidence matrices between r and L, are equivalent, and therefore it follows that these elementary divisors are independent of the chosen ordering on the sets r and 4,. Thus, for our purposes we may speak of the incidence matrix between 4, and 4.

In the cases in which one of the sets is Lo, the SNF is easily computed. In fact, for 0 < r < 2m, the SNF of 77r,0 is the row matrix

(1 0 0 ... 0)

of size 1 x 1ï¿½41.

We now consider the two extreme cases in which one of the sets is Ci. It is easy to see that the SNF of r72m,1 is the column matrix

of size ILWI x 1, while the SNF of rlai is the identity matrix of order ï¿½C11.

In this dissertation, we shall be concerned with the intermediate cases in which one of the sets is Li; i.e., our problem is to determine the elementary divisors of rlr,1 for all 2 < r <, 2m - 1.

1.2 Statement of the Main Result

Denote by Z7r the free abelian group on the set Cr. The incidence matrix between Cj and Cr can be interpreted as the -module homomorphism

?7r,1 : Z *r __ Z ï¿½which sends an isotropic r-space to the (formal) sum of the 1-spaces it contains. Then finding the elementary divisors of the incidence matrix is equivalent to finding a cyclic decomposition of the cokernel of this homomorphism.

Let di be the coefficient of ti(P-1) in the expansion of (z tj) . Note that di is equal to the number of monic monomials in 2m variables which have total degree equal to i(p - 1) and are such that the degree in each variable is at most p - 1, (e.g., see Hamada [18]). Then

di = d2m-i and 2 [- I -i1. (1.1) i=1 P
In (1.1) we are using the p-binomial coefficients

i=1p-

This is the number of s-dimensional subspaces in an n-dimensional vector space over

We now state our main result.

Theorem 1.2.1. Assume 2 <, r < 2m - 1.

(a) Jf r ï¿½ rn, then the incidence matrix between C,. and Ci defines a finite abelian group with cyclic factors of the following orders:

(1) [r] with multiplicity 1;
(2) pr-i with multiplicity di for 1 <, i <, r - 1

(b) The incidence matrix between Cm and Ei defines a free abelian group with free rank equal to p(p- + 1)(pr- - 1) 2(p - 1)

Remark 1.2.2. Replace p by any prime power q (even or odd) in the general setup above, so that V is a non-singular symplectic space over the finite field Fq. Our methods can be used to establish the following results:

(a) For r = m, the only p'-torsion in coker r,1 is a cyclic factor of order [;]q.

(b) We have

rankF 7m7. = C11 - q(qm + 1)(qm-1 1) 2(q- 1)
for any field F whose characteristic is different from p.

For odd q, these assertions can be established simply by replacing p by q in our arguments. For even q, the situation actually simplifies for the following reason: the most difficult part of the p'-torsion in the coker i7r,l for us to compute is the 2-torsion; when q is even, 2-torsion is no longer p'-torsion and we are in fact able to use our methods for handling the odd p'-torsion to handle all of the p'-torsion in this case.

1.3 Background on Incidence Problems

The main problem which we solve in this dissertation (see Theorem 1.2.1) is part of a wider class of problems on incidence. To put our problem in the proper

perspective, we introduce the general notion of an incidence structure and define its associated incidenc( matrix. We then discuss how such a structure may be useful and indicate ways by which it may be studied through its incidence matrix. Finally, we present several examples which are closely related to the topic of this dissertation.

A finite incidence structure is an ordered triple S = (P, B, 1) which consists of two disjoint finite sets P and B, and a subset I of the cartesian product P x B. The elements of P are called points and the elements of B are called blocks. If the ordered pair (p, B) is in 1, then we say that p is incident with B.

Put JPJ = v and IBI = b, and let the points and blocks be labeled as follows: P= {Pi,...,Pv}

and

. =.. . ...,Bb}.

An incidence matrix for the incidence structure S is a v x b matrix A = (aij) such that
I if (pi, Bj) E I
ai~j = if (pi, Bj) _E"

Let F be any field. We denote by CF the column null space of A over F and call it the code over F of the incidence structure. We denote by CF the column space of A over F and call it the orthogonal code over F of the incidence structure. If F has characteristic p, then the dimension of CF- is called the p-rank of A.

In general, the only incidence structures which are either mathematically interesting or practically applicable are those which possess some degree of regularity. One of the broadest classes of such regular incidence structures is the class of tactical configurations (e.g., see Dembowski [10]). These are incidence structures with equally many points on every block and equally many blocks through every point. Many configurations of classical geometry are of this kind (e.g., the configurations of Desargues and Pappus in projective geometry, and those of Miquel and the bundle

theorem in inversive geometry). A tactical configuration with equally many blocks through every two distinct points is called a balanced incomplete block design (BIBD), or design, for short (e.g., see Assmus et al. [2]). These structures are studied not only by combinatorists and coding theorists, but also by statisticians, who use them for the planning and analyzing of agricultural and other experiments.

It is often the case that the most accessible way to study an incidence structure S is through its associated incidence matrix A. For example, an analysis of the three matrices A, AAt, and AtA permits valuable insight into the combinatorial structure of S (e.g., see Dembowski [10], pg. 19-21). Thus, we may sometimes reduce a combinatorial or coding theoretic problem to one about integral matrices. The Smith Normal Form (SNF) has been used extensively in the study of integral matrices (e.g., see Newmann [33]), and so it is not surprising that it has proven effective in studying incidence matrices. We now discuss several ways in which the SNF of an incidence matrix may be of use.

Consider first the common situation of deciding when two combinatorial structures are isomorphic. Often we may view these combinatorial objects as incidence structures, and may therefore consider their corresponding incidence matrices. An isomorphism between the two objects translates to an equivalence of the two incidence matrices by permutation matrices. Since permutation matrices are unimodular (i.e., invertible over the integers), we then see that if the two objects are isomorphic then their associated incidence matrices have the same SNF. We remark that the converse is not true; i.e., nonisomorphic objects may have the same SNF. See Rushanan [34] for an example where two strongly regular graphs associated to latin squares are shown to be nonisomorphic by comparing the invariant factors of their associated incidence matrices.

Another application of the SNF is in computing p-ranks of incidence matrices. Indeed, the p-rank of an incidence matrix A is equal to the number of invariant

factors of A which are not divisible by p. These p-ranks are important to coding theorists. In fact, let IF be a finite field of characteristic p, S a BIBD, A the incidence matrix associated to S, and CFj the corresponding code. Then CF has merit in that it can be decoded by a relatively simple procedure known as majority logic decoding (see Massey [30]). In order for CF to serve effectively as an error-correcting code (see Assmus et al. [2]), it is necessary for the dimension of CF to be large. This is equivalent to requiring the p-rank of A to be small. For an example of the SNF being used to study p-ranks of incidence matrices associated to a class of designs called symmetric designs (i.e., designs in which the number of points is equal to the number of blocks), see Lander [25].

We now discuss several important classes of incidence structures and the progress which has been made on the study of their associated incidence matrices.

Let n be a natural number and put N :.=... , n}. For each I < k < n let Tk be the set of all subsets of N of cardinality k. We consider the incidence structure St,k whose points are the elements of Tk, whose blocks are the elements of T, and where incidence is given by set-theoretic inclusion. The associated incidence matrix will be denoted by At,k. We note that St,k is a tactical configuration, and that, when k = 1, it is actually a design. It has been known for quite some time that At,k has full rank over a field of characteristic zero (see Gottlieb [15]). The 2-rank of At,k was given by Linial and Rothschild [28]. In the same paper, the authors also determined the 3-rank in the special case where t = k + 1. The answer for any prime is given by Wilson [41]. In fact, he determines a diagonal form for At,k which is equivalent to the SNF. Recently, de Caen [7] has derived a recurrence relation for the rank (over most fields) of At,k.

Now let p be any prime number and q = p'. We denote by V an n + 1dimensional vector space over TFq. Then T,. (resp. F,) will denote the set of rdimensional linear (resp. affine) subspaces of V. We consider the incidence structure

S,, (resp. S,,,) whose points are the elements of Tr (resp. '), whose blocks are the elements of T, (resp..Y"), and where incidence is given by inclusion of subspaces. We denote by A,,, (resp. B,,,) the associated incidence matrix. We note that S, (resp. S',) is a tactical configuration and that, when r = 1 (resp. r = 0), it is actually a design. For more information on these structures and for a discussion of how they relate to the well-known Reed-Muller codes, see Assmus et al. [2]. The matrices Ar,i and B,.,1 have been known to have full rank over a field of characteristic zero for some time now (e.g., see Dembowski [10]). This result was generalized by lKantor [23], who showed that whenever I K< r < S < n -r' + (resp. 0 < r < s < n-r) the matrix A,,, (resp. B,,) has full rank over a field of characteristic zero. In fact, it follows immediately from Kantor's arguments that A,,, and B,,, have full rank in characteristic zero for all choices of r and s. The ranks of these matrices in positive characteristic different from p were determined by Frumkin and Yakir [12] (in the linear case) and by Yakir [42] (in the affine case). The p-rank of A,,1 was computed by Graham and MacWilliams [16] in the special case when n = 2, and was then independently obtained by Smith [39], by MacWilliams and Mann [29], and by Goethals and Delsarte [14] for general n. An exact formula for the p-rank of Ari was obtained in the special case t = 1 (i.e., q = p) by Smith [39], and then for all t by Hamada [18]. Hamada [19] gives an exact formula for the p-rank of B,.,1 for all t. We remark that all of these ranks were determined without a knowledge of the SNF.

One of the first results about the Smith Normal Forms for these matrices came in a paper by Black and List [4], where they determined the integral invariants for A,,I in the case when t = 1; i.e., when q = p. This result was generalized by Sin [36], who computed the integral invariants for arbitrary Ar,i (again only when t = 1). Sin generalized the result of Black and List again (see Sin [38]), by computing the integral invariants of A,,, for all t.

Having looked at several related examples, we now restate the main problem of this dissertation in the language developed in this section. Let q = pt be a power of the odd prime p and let V be a non-singular symplectic space of dimension 2m over Fq. Then, for 1 < r < m, C, will denote the set of isotropic subspaces of V and E2m- will denote the set of orthogonal complements of isotropic subspaces of V. We are interested in the incidence structure whose points are the elements of 12i, whose blocks are the elements of 4,, and where incidence is given by inclusion of subspaces. We let rl,, denote the associated incidence matrix. Note that this incidence structure is a tactical configuration which is not a design; this follows from the observation that two 1-spaces are contained in a common isotropic subspace precisely when they are orthogonal to one another. For more information on this structure, see Wan ZheXian [40]. In the special case when m = 2 and t = 1, de Caen and Moorhouse [8] worked out the p-rank of 772,1. This result was then generalized by Sin [37], who found the p-rank of 7r,1 for 1 < r < 2m - I and t = 1. In this dissertation, we first derive those elementary divisors of r7,1 (1 < r < 2rn - 1) which are coprime to p; this result is established for all t. Thus, as a corollary we may deduce the e-ranks of these matrices for all primes ï¿½f p. In the special case when t = 1, we are also able to determine the p-elementary divisors of ?7r,l (1 < r <, 2m - 1). We remark that for r m m the elementary divisors asserted in Theorem 1.2.1 are precisely the same as those found in [36]. This fact can be interpreted in coding theoretic terms as follows: if r : m, then the code coming from the design of 1-spaces vs. arbitrary r-spaces of V is the same as the code coming from the tactical configuration of 1-spaces vs. isotropic r-spaces of V. To actually implement these codes in practice, it is necessary to generate the associated incidence matrices on the computer. Since the number of isotropic r-spaces in V is smaller than the number of arbitrary r-spaces in V, we see that it is computationally more effiecient to implement the (equivalent) code coming from the isotropic subspaces.

1.4 Outline of the Solution

Let G Sp(2m,p) be the symplectic group for V. Then G is the set of all linear transformations from V into itself which leave the form invariant. It follows from Witt's Theorem (e.g., see Artin [1]) that G acts transitively on each of the sets 4, for 1 < r <, 2m - 1. By extending this action linearly ZCr becomes a ZG-permutation module and the incidence map 7r,1 becomes a ZG-module homomorphism.

Let f be any prime. Then fZc4 is a ZG-submodule of Z'r and the quotient Z1,/fZr - FCr is an FeG-module. Since

qrI(4Z') 9 CZ21,

we have an induced FeG-module homomorphism

We call ?,1 the reduction modï¿½ of r,1 and refer to the rank of trj as the ï¿½-rank of 7T,l. Note that the f-rank of 7r,1 is equal to the number of invariant factors of r7r,l which are not divisible by f.

To prove Theorem 1.2.1 we will need to conduct a detailed study of these reduced incidence maps. In order to carry out this study, we will need to know the submodule structure of the permutation module for G on the 1-spaces of its natural module in all characteristics (.

For e not equal to 2 or p, the complete submodule lattice has been determined by Liebeck [27]. We collect these results in the first section of Chapter 2 and then use elementary linear algebra to determine the dimensions of each of the submodules found in Liebeck [27]. In Chapter 3 we derive the modulo 2 submodule structure. The arguments are taken from Lataille et al. [17]. This puts us in a position to completely determine the p'-torsion in coker 77r,1 (see Chapter 4). Although our interest is in the representation theory of G = Sp(2m,p), all of the arguments in Chapters 2, 3, and 4

work with p replaced by q, where q is a power of p. Therefore, in those chapters only we let V denote a non-singular symplectic space of dimension 2m over lFq, and we let G denote the symplectic group Sp(2m, q).

In Chapter 5 we handle the natural characteristic case. To simplify the situation, we localize at p and examine closely the permutation modules and incidence maps over Zp, Qp, and Fp. Here Zp and Qp denote the p-adic integers and the padic numbers, respectively. After first gathering structure theorems from Sin [37], we introduce certain Zp-forms Mr in QpG-modules which are isomorphic (whenever r $m) to the kernel of the augmentation map on Qp'. The submodule structure of the mod p reductions Mr := Mr/pMr of these lattices is the essential ingredient in the determination of the p-torsion in coker rqr,i. CHAPTER 2 REPRESENTATIONS OF G IN ODD CROSS CHARACTERISTIC 2.1 Preliminaries Let q be a power of the odd prime p. Throughout this chapter, V will denote a non-singular symplectic space of dimension 2m over Fq, and G will denote the symplectic group Sp(2m, q). In this chapter we collect results from Liebeck [27] which give the complete submodule lattice of F"C, where F is a field of characteristic not equal to 2 or p. We then use elementary linear algebra to compute the dimensions of each of the submodules found in Liebeck [27]. We begin by establishing some notation and presenting some definitions. It is an easy consequence of Witt's Theorem (e.g., see Artin [1]) that the group G acts transitively with rank 3 on Ai; i.e. there are three orbits for the action of G on pairs of 1-spaces of V. Equivalently, we may say that the stabilizer in G of each 1-space has three orbits on C1. For x E Aj let {x}, A(x), and 4(x) be the orbits of Gx on 1, where Gx denotes the stabilizer in G of x. If we take A(x) = {y E Ai I Y Z x'}, then IA(x)I- qml and IO z l=q(q2m-2_ 1) q-1 and izA(x) n A(y)j q2m-2(q- 1) for x - y in Li with y E A(x).. Denote by F the strongly regular graph on ï¿½1 associated to G in which x is joined to y if and only if y E A(x). Recall that a graph is called strongly regular if all vertices have the same valency and if the number of vertices adjacent to both of two distinct vertices v1 and v2 depends only on whether v, and V2 are adjacent or not. We define the adjacency matrix (call it A) of F as follows: Fix some arbitrary ordering on the elements of L1, so that we may write L - {Xl''-, XlC1l}. Then A = (aij) is the Iï¿½11 x Ii matrix such that ai~j =0 if Xi q A(xj) . Now let F be any field and let IF'- be the associated permutation module for FG. For a subset B C C1, write SB := b FC. bEB In particular, we write There is a natural, non-singular, FG-invariant symmetric bilinear form [-,-] on F" defined by demanding L, be an orthonormal basis, and then extending the action linearly to the whole space. For any subset W C FC' we write W' {:= f vEC I [v,w] =o,Vw E W and call W' the orthogonal complement of W in IF" . Note that we have used the same terminology and notation for orthogonal complements in V, but no confusion should arise since it will always be clear from context which type of complement is intended. We remark also that if W is an FG-submodule of IF", then so is W'. 2.2 The Complete Submodule Lattice Define FG-submodules Uï¿½qm-, of FC as follows: Utqm-i := ((ï¿½qm-x + sA(x)) - (ï¿½qm-ly + sA(,)) I x,y E Cl) where ( } denotes F-span. Following [27] we call U~qm- the graph submodules of FC". We will require the following nontrivial result due to Liebeck [27]: Lemma 2.2.1. (cf. Liebeck [27], Theorem 1). If F is a field of characteristic f, where f 7$ p, then any FG-submodule of f' which is not contained in (1) contains a graph submodule.

Using this result, Liebeck is able to determine the complete submodule lattice of the permutation module for G on the 1-spaces of its natural module in all characteristics other than 2 and p.

Lemma 2.2.2. (cf. Liebeck [27], pg. 10). Assume F has characteristic t, where f = 0 or else char f 7 p is odd.

(a) If f = 0 or if ï¿½ > 0 but f { [27q, then the graph submodules U~qm-i are nonisomorphic, simple FG-modules and we have the orthogonal decomposition

FC'= (1) e U-qr-, e Uqri.

(b) If f> 0 and iffï¿½ qm + 1, then Uq.-i is a simple FG-module, but U-qm-i D (1). The quotient LTqrn-1/ (1) is simple and not isomorphic to ULqm,-. Furthermore , we have

IF1 = ti_ e Uqm-1,

where U_ is a uniserial FG-module which contains U, m-I as a submodule of codimension one. We have

(Uqrn--1) = U-

and

(U_qM-1)' (1) 0 Uq--.

(c) If e > 0 and if e [ T]q then Uqr- is a simple FG-module, but Uqm- D (1). The quotient Uq, .../ (1) is simple and not isomorphic to U-qm- . Furthernore , we have
IK= U+ 0{ Uqm-,

where U+ is a uniserial FG-module which contains U.-i as a submodule of codimension one. We have

(U_qm- ) U+

and
(Uq,,-,)-= ) @U-q--,.

The structural information given in Lemma 2.2.2 will be used heavily in Chapter 4. For certain results we will also need to know the exact dimensions of each of the submodules of FC'. We compute these dimensions in the next section.

2.3 The Dimensions

We now compute the dimensions of the submodules of F" for char F - 2,p. Our strategy is to view the adjacency matrix A for the graph F as an lFG-module endomorphism of F" (in the natural way) and to recognize the direct summands of F" as the generalized eigenspaces for A. We begin with a result from Higman [20]: Lemma 2.3.1. (cf. Higman [20]). If F is an arbitrary field, then the characteristic polynomial of A over F is c(x) = (x - q2r-1)(x + qm-l)'f(x - qm-1)9, where f q(qm-1)(qm-l+1) and g = q(qm+1)(qm--1)
2(q-1) 2(q-1)

Next we will need the following computational results:

Lemma 2.3.2. If x E ï¿½i, then

A(ï¿½qm-l x + sA(x)) = q2m-2(q 1)1 qm-(ï¿½qm-lx + SA(X)). Proof. For x E L, we have
A(ï¿½qm-lx + sz(x)) = ï¿½qr-'A(x) + A(sA(x))

= 5qm-lsA(x) + I sa(y).

But an easy computation shows E sA(y) = q 2m-1l q 2m-2( -x), YEA(X)
and the result follows.

Lemma 2.3.3. We have A2 - q2m-:2 = q2m-2(q - 1)J, where J denotes the all-one matrix. Proof. If x E C1, then A2(x) =A(sA())

YEA(.r)
and so it follows from the proof of Lemma 2.3.2 that A2 = q2m-lj _q2m-2(J - i) = q 2m-2(q_ 1)J + q2m-2j.

With these facts in hand we may compute the dimensions of the FG-submodules of FC whenever the characteristic, f, of F is not equal to 2 or p Proposition 2.3.4. If f :$2 or p, then dimF U-qm- = f and dimF Uqm-i = g. Proof. We proceed by cases. Throughout the following, let K,\ denote the generalized eigenspace for A corresponding to the eigenvalue A. Note that since A is an FGmodule homomorphism, each of its generalized eigenspaces is an FG-submodule of CASE 1: =0. In this case A is diagonalizable (4 is symmetric) and from Lemma 2.3.1 we see that A has minimal polynomial m(x) = (x - q2m-1)(X2 - q2m-2). From Lemma 2.3.2 we deduce U-qm-, C K-qm-1 and Uqm-iCKqm-i. It then follows from Lemma 2.2.2.a that these containments are actually equalities. Thus, dirM U-qm-i = f, and dimF Uqm-, = g. For the remaining cases, assume that f > 0 is odd and unequal to p. CASE 2: f f q2m - 1. The minimal polynomial, m(x), for A over F must divide (x - q2m-1)(X2 - q2m-2). Since q2m-1, ï¿½qm-1 are all distinct modulo f, it follows that m(x) = (x - - q2m-2), and that A is diagonalizable over F. We see from Lemma 2.2.2.a and Lemma 2.3.2 that Kï¿½qmr-i = Uï¿½qm-. As in Case 1, we have diFrnF LT-qm.. f and dinIF UqI .. g. CASE 3: e Iq- 1 bute{ [2eJ] Since _qm-' and qm-l are distinct modulo t, the polynomial x2_-qq2m 2 divides m(x). But by Lemma 2.3.3 we know that m(x) divides x2 - q2m-2. We must then have re~x = 2 -q2m-2, and hence A is diagonalizable over F. Since q2m-1_ qm-1 mod t, we see from Lemma 2.2.2.a and Lemma 2.3.2 that K-qm- = U-qn-1 and Kqm-, i (1) ï¿½ l'q-. Thus, the conclusion follows in this case. CASE 4: e Iq-Iandg f[27] As in Case 3, A is diagonalizable over F with minimal polynomial re~x = 2 - 2m-2. Since t is odd by assumption and since gcd(q- 1, qm + 1) = 2, we must have e [T[r]q Since q2m-1 = qM-i mod t, it follows from Lemma 2.2.2.c and Lemma 2.3.2 that I'qm-1 =. qm-1 and AKq .., U+ So dimF U qM- f and diralF Uq - g. CASE 5: t f q- 1 but I [2,]q The minimal polynomial of A must divide (x - q2m-1)(x2 _q2m-2). 19 Since qr-1 and -q'-1 are distinct modulo f, we see that m(x) is divisible by 2 2m-2 x-q Since t f q - 1, Lemma 2.3.3 shows that =dr) - (x - q2m-1 )(X2 q2 m-2). Since gcd(qm + 1, Itj) = I or 2, we must have f q' + 1 or ( [q If the former holds, then q2m-= _q,,l- mod [, and so Lemma 2.2.2.b and Lemma 2.3.2 show that K-qM-i = Uand Kqm-1 = Uqm-i. Since qm-1 is not repeated as a root of m(x), we see that dirmF Uqm-i = g, the multiplicity of qm-1 as a root of the characteristic polynomial of A. We then deduce dimF U- = 1 + f. On the other hand, if f I q 2m-= qm-l mod f, and then Lemma 2.2.2.c and Lemma 2.3.2 show IKqm-1 t; ...q 1 and Kqrn-r U+. Since _qm-l has multiplicity one as a root of m(x), we see that dimF Uqrn-I f and hence diMFU+ = 1 + g. El CHAPTER 3 REPRESENTATIONS OF G IN CHARACTERISTIC 2 3.1 Preliminaries Let q be a power of the odd prime p. Throughout this chapter V will denote a non-singular symplectic space of dimension 2m over ]Fq, and G will denote the group Sp(2m, q). In this chapter we determine the complete FG-submodule lattice of FC', where F is a field of characteristic 2. The arguments are taken from Lataille et al. [26]. Our approach is to first restrict the action of G to that of a certain maximal subgroup. namely the stabilizer in G of a maximal isotropic subspace of V The composition factors of this restricted action are determined and using a recent result from Guralnick et al. [17], we are then able to determine the composition factors for the action of the full group. This puts us in a position to obtain the submodule lattice (see Theorem 3.6.3). For simplicity, we will always work over an algebraically closed field of characteristic 2, which we denote by k. 3.2 Restricted Action Fix a symplectic basis el,... ,em, fl,..., fm for V over Fq, so that I ifi j (ei,ej) = (fi,fj)= 0 and (ei, (,e) 0 if i " Let M := ,.... ,,) and P := (fi,.... J,,) be maximal isotropic subspaces of V. Let GM denote the set-wise stabilizer of M in G. Then GM = S x L (3.1) where S={(I$) IA =A', A EHon(PM)} (3.2) and

L= {(g 0~t g EGL(M)} (3.3) Here I is the rn x rn identity matrix and 0 is the rU x 'Xm zero matrix.

To determine the kG-composition factors of kL. we will first need to determine the composition factors of Resgu kil, where Res G k"' denotes the kGM-module obtained by restricting the action of G on kc to that of GM. We start by noting that GM has two orbits on ï¿½1 :

and

Now for any subset X C C1, we will let kx denote the k-span of the elements of X. Then we have the following decomposition of ResGM k"' as a direct sum of kGM-submodules:

ResGM kc' = kï¿½l (D k02. (3.4) Thus, to determine the composition factors of ResgM kcI we may separately study the summands in (3.4).

The first summand is easily handled:

For v E V, write v () ,where x EM andy E P. Then

(v) E 02 if and only if y $0. (3.5) With this notation, the computation (IA) (x) =(x + Ay) (3.6) shows that S acts trivially on 01, i.e. S acts trivially on k0'. From (3.1), (3.2). and (3.3) we see that the induced action of GM/S - GL(M) on k"l is the usual action of GL(M) on the 1-spaces of M. Thus, the kGM-submodule lattice of k", is known from Klemm [24]. Explicitly, if we put )7 : w E kï¿½ wEOi and (w - aI w, a E O)k, where ( )k denotes k-span, then we have Lemma 3.2.1. (a) If m is odd, then K is simple and kc = (1C)k - K' (b) If m is even, then k' is uniserial with composition series kOl K (101)k I {o} In the situation of Lemma 3.2.1.b, put 'C' :='C(lo)k. We will indicate the composition factors of kï¿½l informally by writing kOl = k + K if m is odd (3.7) ï¿½(2)k+K' if "t is even Of course, here k denotes the simple trivial module. To determine the kGM-composition factors of the second summand in (3.4), we will once again begin by restricting the action of GM to that of its normal subgroup S; i.e., first we will determine the composition factors of ResGM k0l. We will then use Clifford's theorem (e.g., see Curtis et al. [9] or Feit [11]) along with Lemma 2.1.2 and Proposition 2.2.4 to recover the GM-composition factors. 3.3 The Composition Factors of Ress'M k2 Using elementary linear algebra we see that given any non-zero y E P and any z E M we can always find a symmetric transformation A E Hom(P, M) which sends y to z. Therefore, it follows from (3.6) that the S-orbits on 02 are indexed by the 1-spaces in P. Explicitly, let (yi) ...., ...be a list of the 1-spaces in P. Then the S-orbits on 02 are the sets O(Yi (x) x E M,} Thus, we have the following decomposition of ResGm k02 as a direct sum of kS-submodules: q-1 Res GM k02 k 04 (3.8) i=1 Le .< be the stblze fK(0)) E .S S Y, ( ) E _S The0 k(,) = IndS k, so that by (3.8) we may write q-1 Res GM k =02 Indsk. (3.9) i=l Recall (e.g., see Curtis et al. [9] or Feit [11]) that for an arbitrary group G, a subgroup H G G, and a kH-module M, we denote by IndGM the kG-module induced from the kH-module M. Explicitly, we have IndaGM = kG ï¿½kH M. We now pause to establish a correspondence between the irreducible kScharacters and the symmetric bilinear forms on M. This correspondence will be the key to determining the composition factors of Res GM ko . We start by noting that P _ V/ M = V/ Mï¿½ M, (where M* denotes the dual space of M) so we may identify P with M*. If we also identify M with (M*)*, then we can identify Hom(P, M) with Hom(M*, (M*)*); i.e., we may regard Hom(P, M) as the set of all bilinear forms on M*. The correspondence 0 1 A) -+ A(3.10) then identifies S with the set of symmetric bilinear forms on M*. Under this identification SY, corresponds to the set of all symmetric bilinear forms on M* which have yi in their radical; i.e., Sy corresponds to the symmetric bilinear forms on (Ker yi)*. Now let ( be a primitive p-th root of unity in algebraically closed k. The corresondence f . H _+ (TraCej/jFP(f(-)) (311) allows us to identify the linear functionals on the Fq-vector space S with the irreducible k-characters of the elementary abelian p-group S. Since S is the set of symmetric bilinear forms on M*, we see that S* is the set of symmetric bilinear forms on M. Thus, we may identify the irreducible characters of S with the symmetric bilinear forms on M. Remark 3.3.1. Let N be an irreducible submodule of IndS k and let f E S- be the linear functional which corresponds under (3.11) to the character of N. By Frobenius reciprocity (e.g., see Curtis et al. [9] or Feit [11]), we know that Sy, acts trivially on N. This means that TraceFq/rp(f(A)) = 0 for every A E Sy,, from which it follows that f(A) = 0 for all A E S,,. But as S., is the set of symmetric bilinear forms on (Ker yi)*, this means that the symmetric bilinear form on M which corresponds to f must be isotropic on the hyperplane Ker yi C M. Thus, the irreducible characters in Inds k are the symmetric bilinear forms on M which are isotropic on Ker yi. Again using Frobenius reciprocity, we see that each such form occurs with multiplicity one. In particular, the zero form (which corresponds to the trivial character) occurs exactly once in each Inds k. In fact, it is easily seen that the unique trivial submodule of liid k is where 1w = L k9(y,). (3.13) wEO (YO Now let B be a non-zero symmetric bilinear form on M which has an isotropic hyperplane. Then B has either rank 1 or 2. If B has rank 1, then the radical of B, denoted by Rad B, is the unique isotropic hyperplane for B. If B has rank 2 then M/Rad B is hyperbolic and therefore has precisely two isotropic lines for the form induced from B, i.e. M has precisely two isotropic hyperplanes for B. For all of this linear geometry, see Artin [1]. In light of (3.9), the above then gives us all of the composition factors of Res( k'"2. We record this information as Lemma 3.3.2. Under the identification in (3.11), RessGM k02 has the following composition factors: (a) The zero form, i.e. the trivial character, which occurs with multiplicity g,,,(b) The rank 1 symmetric bilinear forms, where each occurs with multiplicity 1. (c) The rank 2 symmetric bilinear forms having isotropic hyperplane, where each occurs with multiplicty 2. 3.4 The kGM-Composition Factors of k02 We start by examining the S-fixed points of k'2. Define q-I T =O 7 (3.14) i=1 where the T are as in (3.12). Now it is easily seen from (3.1) that GM/S - GL(M) permutes the vectors in (3.13) in the usual way that GL(M) acts on the I-spaces of M*; i.e., in the usual way that GL(M) acts on the hyperplanes of M. Thus, if we let ï¿½m-1 denote the set of hyperplanes in M, then the kGM-module T can be naturally identified with the kGL(M)-module kCm-1. It is well-known (e.g., see Dembowski [10]) that the permutation modules on the 1-spaces and the hyperplanes, respectively, of M are isomorphic over a field of characteristic zero. Therefore, from a general principle of modular representation theory (see Felt [11], Theorem 17.7) we know that V- 1. and k' have the same composition factors. Therefore, it follows from (3.7) that T- = k+K1 if m is odd (3.15) =,(2)k+ Vif m is even We remark here that it can actually be shown that kCm- and kï¿½l are isomorphic for G. To find the remaining composition factors, we now consider the action of GM on the irreducible characters of S. We start by observing that as S acts trivially on its characters, we need only consider the induced action of GM/S - GL(M). Now GL(M) acts by congruence transformations on S. Therefore, if we view the elements of S* as symmetric matrices, then the action of GL(M) is again by congruence transformations. We then see that under correspondence (3.11), GL(M) acts by congruence transformations on the characters of ,. There are two GL(M) congruence classes of rank 1 symmetric bilinear forms, represented by diag(1,O,...,O) (3.16) and diag(a, O, . . . , O) (3.17) where a is a non-square in Fq (see Artin [1]). The stabilizer of both classes is { (:5, 0)} which has index ---' in GL(M). Let B1 denote the congruence class of (3.16) and B,, denote the congruence class of (3.17). Let W1 denote the external direct sum of the S-characters which correspond to the forms in B1, and let W2 denote the external direct sum of the Scharacters which correspond to the forms in B,. Then it follows from Lemma 3.3.2.b and Clifford's Theorem that k12 has composition factors, call them W1 and VV2, which when restricted to S are isomorphic to W1 and W2, respectively. Note that dimk W1 = dimk 2 = q - (3.18) 2 but W1 2 Now, there is one congruence class of rank 2 symmetric bilinear forms having isotropic hyperplane, represented by (1)0 . (3.19) The stabilizer in GL(M) of this class is { 0,' )} where C is a 2 x 2 monomial matrix. This subgroup has index q(qm-)(qm-l-) in GL(M). Let D denote the external direct sum of the S-characters which correspond to these forms. Note that dimk D q(qm-1)(q.-l--) It then follows from Lemma 3.3.2.c 2(q- 1) " and Clifford's Theorem that exactly one of the following cases holds for kï¿½2 : Case A: k'2 has precisely two composition factors, call them D1 and D-1, which when restricted to S are isomorphic to D. Case B: k02 has a single composition factor, call it Do, which when restricted to S is isomorphic to D G D. We now show that the former is true. We start by establishing some notation which we will use throughout the remainder of the paper: For any field IF, we let FCr denote, as usual, the FG-permutation module on 4r, and we let 'lr,s : -I~c be the FG-module homomorphism which sends each isotropic r-space to the (formal) sum of the isotropic s-spaces which are incident with it. We now require some notation which we introduced in ï¿½2.1. For the sake of exposition we repeat the definitions here. Define For each x E 1i put A(x) := {E ï¿½ 2 I Y X 11, and define an element SA(x) E FCï¿½ by SA(x) : X. (3.20) YEA(X) Define a non-singular symmetric bilinear form [-,-]F by demanding that the elements of ï¿½j form an orthonormal basis. For any subset S C FLI put sL := 1v C k" I [v, S]F = 0, for all s E S Note that we have used the same notation for orthogonal complements in V, but no confusion should arise. Note also that if S is a FG-submodule of F', then so is S'. Now let Q2 denote the field of 2-adic numbers and let Q2 be its algebraic closure. Then F will be the maximal unramified extension of Q2 in Q2 (see Iwasawa [22], pg.37), and R will be the valuation ring of F. Note that F has residue field k. From Lemma 2.1.2.a and Proposition 2.2.4 we have that .-c, = (I)T (D U-1 ED U1, (3.21) where Uï¿½r1 are irreducible )7G-submodules with dimT U_-1 =q(q' - 1)(qm-1 + 1) (3.22) 2(q - 1) and dimT U, = q(qm + 1)(qm- - 1) (3.23) 2(q - 1) Let Uï¿½1 be the reductions modulo 2 of U+I. Their restrictions to GM must be collections of the composition factors described above. By (3.22) and (3.23) the dimensions of the composition factors of Uï¿½1 add up to q(qm+1)(q-':lF) Assume now 2(q-1) " that (m,q)$ (2,3). Then
q(qm ï¿½1 1)(qm-' :: 1)
(qm 1)< 2(dimk D). (3.24) 2(q- 1)

So it cannot be that either ResGM U'41 contains a composition factor which when restricted to S is isomorphic to D dï¿½ D. Thus, we deduce that Case A holds. If (m,q) = (2,3), then dimy U-1 = 2(dilnk D). However, it is easy to see (e.g., by considering degrees) that U-1 is the unique non-trivial composition factor which is common to both -C7 and FC2. Since .F2 = IndG MT, it then follows from Frobenius reciprocity that GM (and hence S) has a non-zero fixed point on U-1. This then implies that Resgv U-1 contains a trivial composition factor. Since S has no fixed

points on D D D, we deduce that Case A holds in this case as well. We mention here that it will be shown in ï¿½3.5 (see (3.32)) that D1 - D-1.

Hence, we have found all the kGM-composition factors of k'2. Combining this information with Lemma 3.2.1 and using the informal notation of (3.7) and (3.15), we may now state

Lemma 3.4.1. (a) If m is odd, then

ResgM kc = (2)k + (2)k + W, + W2+ D1 + 'P-1.

(b) If m is even, then

ResGM kL' = (4)k + (2))K' + W1, + W2 + AP1 + 72-.

3.5 The kG-Composition Factors of kcl

Let Y, 7Z, Uï¿½, and Uï¿½1 be as in ï¿½2.3. It follows from (3.24) and the remarks following it that each of Dï¿½1 occurs in exactly one of Resgv U+i, and that the Dï¿½1 do not occur together. Thus, we may assume that our notation is chosen so that DPï¿½1 is a composition factor of ResaM U+j. Also, since q(qm + 1)(qm- - 1) q(qm - 1)(qm- - 1)
2(q- 1) 2(q- 1) qm _ 1
M -1
q -1
qm 1
2
dimk W1 dimk W2,

we deduce upon inspection of Lemma 3.4.1 that

RG f =+ D1 (m odd) (3.25) GesM V =k + V' + D1 (m even) Suppose for the sake of contradiction that U1 has a kG-composition factor, call it K, which when restricted to GM is isomorphic to K. Since S acts trivially on

k, it is contained in the kernel, call it J, of the representation of G on K. Since S is not contained in the center of G, and since the center of G is the only non-trivial normal subgroup of G, we deduce that J must be all of G. But GM acts non-trivially on K;, a contradiction. It follows that U1 has no such composition factor for G, and therefore U1 is irreducible if m is odd. If m is even, then similar reasoning allows us to conclude that either U1 is irreducible or else U1 = k + X, with X irreducible. We now show that the latter is true.
Using the notation in (3.20), we define a kG-module homomorphism

kcl -+ kcl (3.26) by
W - Wo + SA(),

where o (E L1. Put

U := Im 0. (3.27) Now define

U' :(1 -+ U2 I U1, U2 E U)k. (3.28) In the terminology of 2.1 we see that U' is the (unique) graph submodule over k.

It is easily seen that

U, n Rc'

is an R-form of U1 and a pure 7RG-submodule of R". Therefore, U1 n Rc

is a mod 2 reduction of U1 as well as a kG-submodule of k". Since U1 n R, 1 is certainly not contained in (1)k, we see from Lemma 2.1.1 that

U' c U, n TR1,

and therefore U1 contains the composition factors of U'.

We require the following result:

Lemma 3.5.1. If m is even, then (1)k C U'. Proof. Let M E Lm. Then an easy computation shows that

Y, +t; 1. (3.29) WCM
oE 1

Now the number of 1-spaces in M is Qm- which is an even number since m is even.
q-1

Thus we may group the summands in the left-hand side of (3.29) into pairs. The result then follows from the definition (3.28) of U'.

[]

Since U1 n Rci has at most 2 composition factors, it follows from Lemma 3.5.1 that

U, i Zc = U'.

We may summarize the above as

Lemma 3.5.2. U1 and U' have the same composition factors.

(a) If m is odd, then U' is simple.

(b) If m is even, then U' is uniserial with composition series U'

(1)k

{0}

In the situation of Lemma 3.5.2.b, we put U'/ (1)k := X. In the situation of Lemma 3.5.2.a, we will use X' to denote the composition factor isomorphic to U'.
In view of Lemma 3.5.2, we have only to determine the composition factors of U'_1. We do this now:

A simple matrix computation shows that 1P2 = 0.

Thus,

U C Ker (p, where U is as in (3.27).

Since 9 is symmetric, we see that Ker o = U'. It then follows that U - kC/Ker o= kcL/Uï¿½ - U*,

(3.30)

(3.31)

i.e. U is self-dual. So from the structure of U' given in Lemma 3.5.2 we deduce Lemma 3.5.3. (a) If m is odd, then U = (1)k E) U'.

(b) If m is even, then U is uniserial with composition series

U

(1)k
I
{0}

We next observe that U-1 has the same composition factors as k'c/U. But since U C U', and since k''/I, - U, we see that kl/LJ contains the composition factors of U. We pause now to note that this implies that D, is a composition factor of ResgM U-1. Since D-1 is the only composition factor of Resgm U-1 of the same dimension as DI, we deduce that

(3.32)

D1 - D)-1,

as was promised in ï¿½3.4.

It remains to determine the kG-composition factors of U'/U. By inspecting Lemma 3.4.1, we see that

ResGM U'iU = W1 + W2.

We now show that U-1 has kG-composition factors, call them W and W2. which when restricted to GM are isomorphic to 1W1 and W2. We will need to consider the conformal symplectic group

CSp(2m, q) :={T E GL(V) I I a E EF so that (Tv, Tw) = a(v, w), Vv, w C V}. For brevity, we put F := CSp(2m, q). Then F G x i] and it is easy to see that U is a module for F. Therefore, U'/U is also a kF-module.

We claim that U'-/U is simple for F. Suppose not. Then it follows that U'/U has kF-composition factors, call them 1, and 042, which when restricted to GM are isomorphic to VV1 and VV2, respectively, and (hence) when restricted to 5 are isomorphic to W1 and W2, respectively. Since W, and W2 are not isomorphic as k'-modules, we see that the following result then leads to a contradiction: Lemma 3.5.4. The kS-modules W and W2 are conjugate for F. Proof Let j3 E Fq be a non-square and consider the element E F whose matrix representation with respect to the basis in ï¿½2.1 is g 3 0- (E ~ Nt (S),

where Nr(S) denotes the normalizer of S in F. If h ( A) E S, then an easy computation shows that
(hO_,= (i ),

i.e. g acts as multiplication by /3 on S. It then follows that acts (on the left) on S* as multiplication by i3-1. Under the identification in (3.11), this means that , acts as

multiplication by 3-' on the characters of '. Taking 3 = a, where a is as in (3.17), it is now easy to see that the conjugate by of the form in (3.17) is the form in (3.16). The result now follows from the construction of the Wi's in ï¿½2.3. Li

Thus, U'/U is simple for F, and it follows from Clifford's theorem that U'/U is semi-simple for G. Now, either U'/U is a simple kG-module, or else U'/U W1 E W2, where WI and W2 are simple kG-modules which when restricted to GM are isomorphic to W1 and W2, respectively.

Consider the following result from Guralnick et al. [171: Lemma 3.5.5. (cf. Guralnick et al. [17], Theorem 2.1) Any irreducible kG-module of dimension less than (q' - 1)(qm-q) is either the trivial (q(m+1)
module, or a module of dimer ion (q2ï¿½I)
2
An irreducible kG-module of dimension (qï¿½I will be called a Weil module (see Guralnick et al. [17]).

This result shows that we must have U'/U _ W1 E W2, where W1 and W2 are irreducible Well modules of dimension @f-2-. Hence, we now have all of the kGcomposition factors of kcl.

If we let V1 and V2 be submodules such that U C V1,V2 C U' and V1/U -W and V2/U - W2, then the above arguments yield the following filtration of kcl

I
I J

Vl V2 (3.33)

U
I
{Of

3.6 The kG-Submodule Lattice of kc

By the minimality of U' (see Lemma 2.1.1) it suffices to determine the submodule structure of (U')'/U'. We start by defining submodules C and C+ as follows: C := Im 7m,1

and
C+:=(x + y I x, y EC')k

We will need the following results: Lemma 3.6.1. (a) C+ C C

(b) Homka(kCm, W) = {0}, for i = 1,2.

(c) C has no quotient isomorphic to W, for i = 1, 2.

(d) HomkG(kCm, k) - k

(e) C+ is the unique maximal submodule of C. Proof. An easy computation shows that [x + SA(x), ?7m,(M)]k = 1, for all x E Cj and all M E L,. From the definition of C+, we then deduce that C' C (C+)". Now (a) follows by taking orthogonal complements.

We have RessG Wi - Wi, for i = 1,2. But from our construction of the Wi in ï¿½2.3, we know that they are fixed point free for S. Therefore, GM has no fixed points on the ResGM Wi - Wi; i.e., HomkGM (k, ResGaM Wi) = {0}, for i = 1,2. Since
kCm = IndGM k,

the assertions in (b) follow from Frobenius reciprocity. Since C is a homomorphic image of km, we see that (c) is an immediate consequence of (b).

Again by Frobenius reciprocity, we have

HomkG(km, k) _ HoMkGM(k, k) _ k. This proves part (d).

It follows from (a) and (d) that C+ is the unique maximal submodule of C with trivial quotient. From Lemma 2.1.1 we have U' C C. Using the inner product computation at the start of the proof, we have C C (U')'. Thus, U' CC C (U')'.

Since

(U')"'/(U') = k + k + W, + W2, we know that any maximal submodule of C with non-trivial quotient must have quotient, W, or W2, which is impossible by (c). Then (e) follows.

Since C+ is not orthogonal to C, we get CnC' C C,

and hence

CnC" C C+.

by Lemma 3.6.1.e. Thus, the quotient C/(C n C') has at least 2 composition factors. Furthermore, C/(C n C') has a unique maximal submodule, namely C+/(C n C'). Lemma 3.6.2. (a) Ci(C q C) is self-dual.

(b) C/(C n C') has a unique maximal submodule and a unique simple submodule. Both the head and socle of C/(C A C') are trivial. Proof. The form induced by [-, -1k on the quotient C/(C nl C) is non-singular and therefore induces an isomorphism between C/(CfnCï¿½) and its dual. Since the form is

G-invariant, this is actually a kG-isomorphism, and (a) follows. Part (b) then follows immediately from the remarks following Lemma 3.6.1.
Fl

In light of Lemma 3.5.4, it follows from Clifford's theorem that any kr-module having at least one of the Wi as a composition factor for G must have the other as well. Since C' and ("I are modules for 1'. we deduce fr-om Lemma 3.6.2 that either

C(C n C') = k + k (3.34) or

C/(C n CI) = k + k + W, + W2. (3.35) Suppose by way of contradiction that (3.34) holds. By Lemma 3.6.2.b it must then be the case that C(C n C') is uniserial. But as G is perfect, it has no module which is a non-split extension of the simple trivial module by itself. So (3.35) holds and it follows that C = (U')' and C+ = U'.

We may now state our main result:

Theorem 3.6.3. Using the above notation, k has the Jollowing submodule lattice:

39

m even m odd

k
(1)' A

x C (1)'

C+ C+

Ik (C+)

xC
(1)i
k \

{o} {o}

Proof. Let N be a kG-submodule of kC. Assume N 0 {} or (1). Then we know from Lemma 2.1.1 that U' C N. If we assume that N 4 kc or (1)', then we have that N' : {0} or (1). But then from Lemma 2.1.1 wehave U' C N'; i.e.,N C (U')'. Thus,

U' C N C (U')'.

From the remarks immediately following Lemma 3.6.2, we know that U' = C' and U' = C+. Thus, if N $U' or (U')', it follows from Lemma 3.6.2.b that U C N C U'. But as U'/U -_ W (j W2, and since W, 9 W2. we see that V and V are the only kG-submodules between U and U' i.e., N = V or V2. 11 Although the dimensions of the submodules pictured above have been given earlier, for convenience we recall here that q(qm - 1)(qm- + 1) 2(q- 1) and diZM k V1 = d irnk V2 - q 2m _l 2(q 1) Retaik J.6.4. Froin Guralnick et a]. [17] wc know that the Weil inudule,, ca 1)(realized over F2 if and only if q ï¿½- 1 mod 8. If q ï¿½- 3 mod 8, Then the smallest field of definition for the Weil modules is F4. With this insight, we may deduce from Theorem 3.6.3 the complete FG-submodule lattice of IF"' for any field F of characteristic 2. Explicitly, if q =ï¿½ 1 mod 8 and F is arbitrary, or if q ï¿½= 3 mod 8 and F4 C F, then the submodule lattice of F'" is as pictured in Theorem 3.6.3. However, if q =ï¿½ ï¿½3 mod 8 and F4 Z F, then the submodule lattice is as pictured in Theorem 3.6.3 except that the quotient C+/(C+)' is irreducible. CHAPTER 4 DETERMINATION OF HE p'-IORSION 4.1 Preliminaries As in the preceding two chapters, we let q be a power of the odd prime p, and V be a non-singular symplectic space of dimension 2m over IFq. We denote by G the associated symplectic group Sp(2m, q). For any field IF, we let 7,I : IFC' C denote, as usual, the incidence map between IF', and Fc". Also, we let : - IF be the FG-module homomorphism which sends each R C C,. to 1 C IF. We call E,. the augmentation map of IF',-. In the case where r = 1 we shall simply write E instead of g Note that er may be identified with the incidence map 71,,o IFC, - FCo. We present two lemmas, the proofs of which are easy computations: Lemma 4.1.1. Assume 2 <, r <, m. If x E Lj and R E C,, then ï¿½qm-, if x C R q x +RSAqmx 0 if x C R',x Z R qr-1 if x Z R' Lemma 4.1.2. Assume 1 < r These simple computations yield the following useful result. Proposition 4.1.3. Assume IF is a field of characteristic [$ p.

(a) For r : we have ,l(Ker 6r) )f Uqm-i.
(b) If f 7 2 we have ImF '7m,F ) U_qm-I but ImF i'm,1 1 Uqm-i.
(c) If = 2, thenUqm-1 Uqm-, and IMF i ,F I Uq-1. Proof. Let X1, X2,. . . ,m, Y1, Y2, . ,ym be a symplectic basis for V, so that

(xi, xj) = (Yi, Yj) = 0 and (xi, Yj) = fi j

Assume that 1 < r < m - 1 and put R1 := (Xi,... Xr) and R2 (X2 .... Xr+i). Then R1, R2 E Lr. If 2 < r < m - 1, then using Lemmas 4.1.1 and 4.1.2 we see that ['ri(/ i) - rAl(/R2), (ï¿½qm-1 (X2) + SA((12 )) - (ï¿½qm-1 (Yi) + =(( qr-1 0, and
-ï¿½ ?7- (yj -s (y)) =2. 0
[)ir,1(R) -?rjl(R2), (ï¿½qm- (X2) + SA(('2})) - (ï¿½qm- (Y) - Sa((1)))] -qm1 ) 0, which establishes the assertion in part (a) for these cases. If r = 1, then [?r,i(Ri L) - r,(R'), (+qm-1 (Y2) + 'SA((y2)))- (ï¿½qm- (yl) + s((q)))] 2q22 # 0. proving the assertion in part (a) for this case as long as f 7 2.
Suppose f = 2. Then there is a unique graph submodule, call it IT,, and by Lemma 4.1.2 we see that

[O.,(R ),((Xr >I - SA()))- ((X2) +t IX(: >) :/- 0. This shows that

Im 2m-1,1 4. U1. (4.1) Since [2,l-1]q is odd, we see that

Im 172m-1,1 Z Ker E.

(4.2)

Then equations (4.1) and (4.2) imply that

Im 72m-l,1 - FCï¿½ (4.3) To see this, suppose for the sake of contradiction that h1 q2n-I.1 C IFI . Then by (3) we deduce that (IM ij2,1,1 ' is a non-zero subimodile of F" not contained in (1) So by Lemma 2.2.1 it must contain U1, contradicting (2).

But then (4) implies i/2m-i,l(Ker Er) = Ker g and part (a) is then seen to be true in this case as well.

To prove parts (b) and (c), let M E Cm, so that M = M'. We see from Lemma 4.1.1 that

[rm,i(M), (qm-lx + SA(x)) - (qm-ly + sA(y))] = 0 for all x, y C C1, but

[ilni(M), (-q 'f-l"x + 6sL(x)) - (-q '(-ly + sA(y))] = -2qT 1if x C M and y Z M. Parts (b) and (c) now follow.

We now deduce an important characterization of r,,(Ker E,.).

Proposition 4.1.4. If F is a field of characteristic f $p and if r m, then qlj(Ker &r) = Ker g. Proof. We know that qi,,(Ker er) C Ker E. Suppose this containment were strict. Then we would have (Ker r)' C (fij,,(Ker ,)'. But (Ker s)' - (1). so Lemma 2.2.1 then says that (r,.I(Ker gi))ï¿½ contains a graph submodule. By Proposition 4.1.3 this is impossible. LI We finish this section with the following: Proposition 4.1.5. Let F be a field of characteristic f # 2,p. (a) Ife=0 or if te>O bute{ [2-] then Imp- m,1 =(1)+ Uqm-i. (b) If e >0 and ï¿½ qm + 1, then ImF m.1 i U-. (c) J f >0 and ( [] thn ImF m,1 = (1) & Uq. Proof. We begin by noting that ImF 'm,i is a nontrivial FG-submodule of I"c' which is contained in (Uqm-)', by Proposition 4.1.3. Suppose first that the hypotheses in (a) hold. Since IMF , 0,1 Z Ker g , the assertion in (a) follows immediately from Lemma 2.2.2.a. Under the hypotheses in (b), we see from Lemma 2.2.2.b that either or Im' 7mF U qm-1. But since e f [T ]q, we see that I F m jr , K er g and the result follows. Finally, if t [[1]q, then Lemma 2.2.2.c shows that either IMF m1={)QU_-or Now given M G fm we see that A(qm,i(M)) = qm-l(1 - lM )), showing that ImF ,1 K-qm-U-qm-1 (see Cases 4 and 5 in the proof of Proposition 2.3.4), so that we must have ImF?7in,1 (1)e Uqm-l. 4.2 Determination of the p/-torsion in coker ,r,1 Let r,: Z', -4 ZC" and E, : Z -, + Z denote the incidence and augmentation maps, respectively, over Z. When r = 1 we shall simply write g instead of gi. We may now prove Proposition 4.2.1. If r = m, then the only p'-torsion in coker 7r,1 is a cyclic factor of order Proof. We have F(ZC1) = z and -(hm lJr) = 1 . and so Z-'/(h - ,, + Ker ) -- Z/ q (4.1) Since r - m we see from Lemma 2.2.2.a and Proposition 4.1.3 that 7r,j Is surjective over a field of characteristic zero. Thus, Imz 77r,, has full rank, and so coker 71,l is a finite group. Therefore, from (4.4) we deduce Icoker 7r,11 = I(Im r,l + Ker E)/Im 7r,i X (4.5) So we must show that Ker E/(Ker e l Im ?rr,I) is a p-group. Let ï¿½ be a prime different from p. Since r(Ker 6,) = Ker - n Im r, we need to show that jr,I(Ier 6r) = Ker E, where Ker Er := Ker er/fKer Er and ,1 is the reduction mod f of 7r,. Since Ker Cr is a pure submodule of ZC,, we may identify Ker Er = Ker Er/tKer Er = Ker r/(Ker 6r n lï¿½Cr) with the image of Ker E, in Zr/Le2Cr . which is (Ker g, + fZï¿½L)/fZeL' Eher , where 4r denotes the reduction mod C of Er Thus, we need to show that r,l (Ker Er) Ker 6. But this last fact has been shown in Proposition 4.1.4. Taking q = p in Proposition 4.2.1, we see that Theorem 1.2.1.a.1 has been established. We conclude this section with the following: Proposition 4.2.2. Asstum K # p i. priiu Tht t coker i1m,1 has ao -torsioli. Proof. We must show that dimF, Im r/m,i = rankz Im 7m,1, for all primes ï¿½ p. If t is odd, this follows from Proposition 4.1.5 along with Proposition 2.3.4. If f = 2 then this follows from Theorem 3.6.3. CHAPTER 5 DETERMINATION OF THE p-TORSION 5.1 Preliminaries Throughout this chapter V will be a non-singular symplectic space of dimension 2m over IFp. We denote by G the associated symplectic group Sp(2m, p). In light of Propositions 4.2.1 and 4.2.2 we are have only to determine the p-elementary divisors in order to prove Therem 1.2.1. Thus, we shall simplify the situation by localizing at p. Let Zp denote the ring of p-adic integers and Qp the field of p-adic numbers. Let rs ï¿½Zr -4 Z and denote the incidence and augmentation maps, respectively, over Zp, and let and P p denote their reductions mod p. Put =. =N r E, For a ZPG-module A we shall use the notations QpA Qp ï¿½z1P A and A:- A/pA. Since p { ICr1, we have the splitting ZC =2P1 eD Y, and hence also F"1=Fp1 a)r. It is easily seen that and consequently ,~ ) C V;. Since the p-torsion for r,. comes from its action on .. we will restrict our attention to these submodules, and shall use the same notation for the restricted maps. We will sometimes need to consider the incidence maps 7lr,s between all the r-spaces of V and all the s-subspaces of V. 5.2 Related Structure Theorems We begin by collecting several results from Sin [37]. The first result gives the submodule structure of V1. Lemma 5.2.1. (cf. Sin [37], Theorem 1). The 1FpG-module Y1 has the following submodule lattice: Wi I W2 Wm W+ W Wm+I 14 2 i-2 tf W42m- 1 {0} For i$ m, the quotients Si :Wi/W,+ are simple. The quotient Wm/Wm+i is the direct sum of two non-isomorphic simple modules, call them S+ and S. Our notation is chosen so that W+/Wm+i :- S+ and W-/Wm+i - SFrom Sin [37] we also have that

dimFp Si = di for i r m, (5.1)

dirmF, S+ + dirnIF. S- din, (5.2)

dinF, S+ - dimrF 5 p' (5.3) and

Si - S2m-i as FG-modules.

(5.4)

Now Cj and L2,-1 are isomorphic as G-sets, via the map which sends each 1space to its orthogonal complement. Thus, the permutation modules Y1 and Y2m-1 are isomorphic, and it follows from Lemma 5.2.1 that 1Y2m-1 has the following structure:

2n
11
U2m-i
I
'2m-2

Urn

I

U2
I

U1
Uo1

Here we have Ui/Ui-1 " Si for i rn and UnL/U-1 5- 5+ ï¿½ S'-. Our notation is chosen so that U+/U-j - S+ and U-/1/,-m - S_.
We need in addition the following fact.

Lemma 5.2.2. (cf. Sin [37], Lemma 4). Yr, has a unique maximal submodule with simple quotient isomorphic to Sr if r 7 m, and S+ if r = m.

5.3 The Modules Mr

Let Mr := Y/Ker r1r,1. We have M1 = Y1 and M2m-1 = Y2,-1. Since Ker 77r, is pure, Mr is a Zg-form in the QpG-module QpYr/QpKer r,.. When r 7 m, we have

QpYr/QpKer ?r,1 - Qpyi.

It follows from a general principle of modular representation theory (e.g., see Feit [11], Theorem 17.7) that in this case all of the mood p reductions M, have the same composition factors as Y'7 and !Vm-i.

The following simple observation will be used frequently. Let 1 t <, s < r < m. Then it is easy to check that 77St o 7r,s = [r t] r,t (5.5) S tP

It then follows that 7r,,s(Ker i7r,l) C Ker 71,1. Thus, we have induced ZPG-homomorphisms ,u. Mr -+ Ms

and

A,, Mr-+ 4l

along with the relations

I/st 0/r,s = r t , /Ir,t and /s,t 0 tr,s = t p /Ar,t (5.6) Lemma 5.3.1. For 2 <, r < 2m - 1 we have Ker 7r,1 = Ker 77r,2m-. Proof. We know from Sin [36] that Ker r, = Ker r,2m-1. Since Yr C Y r, and since T7r,1 and r/r,2m-1 are just the restrictions of ir,1 and i,,2._ to Y,., we have Ker 7r,1 = Ker fir,j n Yr = xKer /r,2m-1 n Y'r = Ix'r 11r,2m-1.

From Lemunia 5.3.1 it follows that we have induced maps

,2m- : M. --+ IY2m-1

and

JAr,2m-1 : ir -+ 2m-1. Lemma 5.3.2. For 2 < t < s < r <, m, we have the relation 7s,r 0 7t,s = [-]plt,r. Proof. This follows by taking transposes in (5.5).
L

Lemma 5.3.3. For 1 < s < r < m 1. we have 'rj,,,.(Ker 77,j) C tNer ,l. Proof We know fromi Sin [36] that

so that
7), (Ier ,2 -)C Ker 7),.,2m-l. Restricting 7,r, 7s,2m-1, and 77,2m-1 to Y, Y8, and Yr, respectively, gives ?7,r(Ker ?7s,2m-1) C Ker r7r,2m-1. The result then follows from Lemma 5.3.1.

From Lemma 5.3.3 we see that there are induced maps Ils,,. M3 M,.

and

As,r Ms M Air,

for 1 ,s
The next result gives the submodule structure of Mr. Lemma 5.3.4. Assume 2 < r <, m- 1

(a) Mr has a unique maximal submodule with simple quotient isomorphic to Sr.

(b) The maximal submodule of MI is the direct sum of a uniserial module Jr Ker / ,j, which maps isomorphically under Pr,2m-1 to Uri-, and a uniserial module J = Ker ir,2r -1, which maps isomorphically under i'r.1 to Wr+i.

Proof. As Mr is a homomorphic image of Y'., the property in part (a) follows from Lemma 5.2.2. To prove (b), we observe that by part (a) the image of the non-zero homomorphism /r,1 in 1 must be either Wr or W2m-r. But the dimension of Im A,, is equal to the p-rank of r,1, which by Sin [37] is equal to diMF, Wr > dim W2m-r. So Im j =,1 Wr. Thus, the kernel J7 has composition factors S1,..., Sr-1. Similarly, Im /r,2m-1 equals Ur or U2m-r. But since Ker 77r,1 = Ker ?r,2m-1, we see that the dimension of the image of Pr,2m-1 is equal to the p-rank of rr,2m-1, which is equal to the p-rank of r72m-r,1. It is shown in Sin [37] that this common rank is equal to dim F, Ur < diMFn U2,,-r. So Im /P,2,-1 = U,. Thus, the kernel J+ has composition factors 5,.+ 1,. ... ,m- 1, ! +, S-.. . .q.m+l ....q 2, 1.

We claim that J,7 n Jj = {0}. In fact, if r im - 1, let N be the smallest submodule of Mr with two composition factors isomorphic to Sr+i - S2m-r-l, and if r = rn - 1, then take N to be the smallest submodule of Mr with composition factors isomorphic to both S+ and S_. Then the above remarks show N C Jt, so that dirnFN < dinFJ,. But by definition of N, we must have /r, (N) = Wr+i. Since dimF Wr+x = dimFJ, we then have dimFN >, din FJ+. It follows that N = J+ and that ArI is injective on N, i.e. J+ n Jr- = {O}. Hence, each map is injective when restricted to the kernel of the other. So Jr- is isomorphic to a submodule of Ur, which from inspection of composition factors must be Ur-i. Also, as shown above we have J+ - Wr+l.
ED

Lemma 5.3.5. For 2 < r

Proof. Suppose for the sake of contradiction that Im /Zrr-1 J -,. Then using Lemma 5.2.2 we see that A,-1,2m-1 (Im lt,, -) is either U, or U2,-,. But from Lemma 5.3.4 we know that Im Pr-1,2m-1 = Ur-1. Since Ur-1 C Ur C U2m-r, we have obtained a contradiction. So Im Pr,-1i C J+1. Again using Lemma 5.2.2 we then see that the image of ,,r-1 is either J,+l, or the proper submodule of g+_1 which has {S2m-,.,... , S2m-1} as its set of composition factors. By (5.5), we have

P ,r . = - P 2 ,1 0 . . . 0 P r - 2 r - 1 0 M r .- - , ( 5 . 7 )

up to a non-zero scalar in lF.. It follows that the rank of P,, is not more than the rank
--2m- Ii and of/,,1. Since, by Lemma 5.3.4, the former map has rank equal to z.,i=,. d since -2m-1 di = dimF J+ we must have Im Yrr-t = J-l. But (5.7) also shows Ker P,r-1 C Jr-. Since dirnFp Ker i,,._i = P__-- di, this containment must actually be equality, and the assertion is proved.

Define []21 to be the number of r-dimensional isotropic subspaces of a non-singular 2m-dimensional symplectic vector space over Fp. Explicity,
r-1
[2mn] = 11(pM-i+l1) [m] I. r iso i=0

We will need the following computation.

Lemma 5.3.6. Assume 2 < r < n -1. Thn the (igivaliuts of , 01/i" ZL' -- Z" are
(i) [ ] with multiplicity 1.
r-2 sno
(ii) pr- r2m-4] (pmr- + 1)(pm-1 - 1) with multiplicity f,
Lr-2 ]iso pr -1 __ 1

(iii) pr-- [2m-4] (p-,. - 1)(M-1 + 1) with multiplicity g. Proof. A straightforward computation shows that

2rr-,r [ -2 1 + [2r - 4](-A-I)
T~,10 ~l I r -1 Lo r-2iso

Since I, J, and A are commuting, diagonalizable transformations, we may simultaneously diagonalize them. But J has rank equal to one with 1 as eigenvector. The result then follows from Leriinia 2.:3.1. El Corollary 5.3.7. If 2 < r < in - 1, then Im A,, o ',,= {0}. Proof. This is immediate from Lemma 5.3.6.
El

Lemma 5.3.8. If2 < r < 77 - 1, then v, (det(u,-i 0 r-i,r)) 2jip 1. Proof. If r = 2, the claim is seen to be true by Lemma 5.3.6. Assume that the claim is true for all numbers smaller than r, and put

C := Pf2,1 0 P3,2 0 ... 0 -tr,r-1 0 Pr-i,r 0 ..0 /2,30 P1,2" Then we see that
1'
vp(det(C)) =Z {vp(det(ai_.,-t )) + vp(det(yj,-i_ ))}
i=2

2- ~ [ -I) + vp (det(lrx-1 0 P-~)
i=2 P

by induction. But using (5.5) along with Lemmas 5.3.2 and 5.3.3, we see that C = Pj,1 0 11,r up to a unit in Zp. Thus, by Lemma 5.3.6, we have

vn(det(C)) = (r - 1)mI
and the assertion follows.

Corollary 5.3.9. If2 < r rm - 1, then Im Pr lr,, # {0}.

Proof. If the linear transformation Kir-i,r were identically zero, we would have vp(det(ttrlr)) 1 [2m] -1

- dimnFp Mr-1.

But from Lemma 5.3.5 we know that
r-1

dimFp J,

so that
Vp(dt( ,_ .,)) + I~(dt(jt,.., ) >+ - 1 + di

> --w1 1,
I P

which is impossible by Lemma 5.3.8. So Im Ar-l,r is a non-zero submodule of Mr.
0l

Lemma 5.3.10. For 2 <, r < m - 1, the following hold:

(a) Im -r =

(b) Ker 1'r_,, = J+J,

(c) Im /11,r is isomorphic to S1 and is contained in Jr-. Proof. We proceed by induction. If r = 2, then by Corollary 5.3.9 we have 1m/1,2 {0}. By Corollary 5.3.7 we have liii pj,2 C J-. and the assertion is seen to be true in this case.

Assume the assertion is true for all numbers smaller than r. where r > 3. If Im /r 1,r Z Jr-, then from Lemma 5.2.2 we see that /r,1 (Im /-x,r) is either Wr-_ or W2,-,.+1. But from Lemma 4.6 we know that Im btr,1 = Wr. Since Wr C W,-,, we must have fr,1 (Im /l--,r) = W2,-r+1. It follows that Ker IXr,1 oIr1,r has composition factors Si,.. , S2m-r. From the characterization of J+t_ in the proof of Lemma 5.3.4 we then see that Ker irj o r- 1,r = J+1. Now from Corollary 5.3.7 we know that the

composition yr,1 o pl is identically zero. But using (5.5) along with Lemmas 5.3.2 and 5.3.3, we get

Yr ,10 /'l,r = 'r,10 i -lr 0 jl,r-1

up to a non-zero scalar in lF. Thus, Im p I,-I C Ker l0 /r 7+1., = , contradicting the induction hypothesis. So Im /1,.-,,. is a (tion-zero) submodule of J,- which, by Lemma 5.3.4, has a unique maximal subinodule with simple quotient isomorphic to Sr-1. It must be that Im ,r-l,r = J,-, establishing part (a). We then see that Ker/Pr-ir has composition factors Sr, ,.., S2m-1. From the characterization of J+1 in the proof of Lemma 5.3.4 we deduce Ker /r-r1,r= J+'_l, establishing part (b).

It remains to prove that Im i'1, C Jr and that Im /t,, S1. But Al,r = 1r-l,r 0 Al,r-l,

up to a non-zero scalar in Fp, and from the induction hypothesis we get hn ftI,,-I C J,7 1 and In /1,_-1 I >1

Since J,7_, I n/,+ = {O} and since J, = Ker fi,,., it follows that tm Pir is a non-zero submodule of J- = hn S ince any non-zero homomorphic image of Yj must have a unique maximal submodule with simple quotient isomorphic to Si, we see that part (c) is established.
LI

5.4 Determination of the p-torsion in coker Ur ,

Proposition 5.4.1. Assume 1 <, r < m - 1. The p-elementary divisors of the incidence matrix between Cr and ï¿½j are pr-i with multiplicity di for 1 < i <, r - 1.

Proof The image of the incidetice map, 'i,. between 1-spaces and all r-subspaces of V contains Imz 1r,1, and hence coker i) ,, is a homomorphic image of coker 71.,1.

Since the asserted elementary divisors are the same as those of 7r,i (see Sin [36]),
r-I
and since from Sin [36] we have vp(Icoker ir,1I) = 3 (r - i)di, it will suffice to show
i-i
r-I
vp(det(=r)) (r - i)di.

For 2 < s ( r, Lemma 5.3.10 shows that Ker 1s = -, so that

2m-1
Vp(det(yI-tsl,)) >/ di t=s
= diMF, J+-,

and Lemma 5.3.5 shows that Xer J/1,s-1 , so that

i=1

tip~d=tdits'j )]di

But then Lemma 5.3.8 shows that we must actually have equality in both places above, i.e.
s-1

i=1
and
2m-1
vp(det( ,,i,)) = 3 di
i=s
for all s. But then (5.5) implies

vp(det(p,,,)) = 3 v(det(y ,-,))
r s1

S=2 i=1

= > (r- di

and the result is proved.

Since for x E Cj and R E Cr we have x C R if and only if R' C x', we see that Proposition 5.4.1 gives the p-elementary divisors of the incidence maps rT2m-r,2m-I, for

2 < r < r - 1. Note also that since ï¿½2m-1 is precisely the set of all hyperplanes of V, the elementary divisors of 172m-1,1 are known from Sin [36]. With these observations we may prove

Proposition 5.4.2. Assumi 1 <, r < in - 1. ihen th( p-clementary divisors of the incidence matrix between C2,-,. and ï¿½ are p2m-r-i with multiplicity di for I < I 2n - r - 1.

Proof. It is easy to check that

772m-1,1 0 12m-r,2m-1 = p 7]2m-r,1 + 1 1 J

But then the induced maps satisfy

r-1
/12m-1,1 0 /12m-r,2m-1 = p /12m-r,l, since J 0 on Y2m-r = Ker J. So, if we put n:= ([271 _ 1), then

det(/t2,-1,1 )det(/12,n-,. 2,yi-1) = p (r- 1)?det(P2m-Y,,l),

and so
1-I 2m-2 2m-1 vp(det(i-2m.-",)) ((r - i)di + Z (2m -1-i)dj - (r- 1) di

2m-r-1
- > (2m -r-i)di.
i=1I

But the incidence map, i)2m-r,1, between 1-spaces and all (2m - r)-subspaces of V, is such that
2m-r-1
vp(lcoker '2,m-,.=j1> (2m - r- i,
i-i

by [36]. The result then follows as in the proof of Proposition 5.4.1.

From Lemma 5.2.2 we see that ImFp rem,1 is a submodule of Yi which has a unique maximal submodule with simple quotient isomorphic to 5+. It then follows

from Lemma 5.2.1 that ImF, = H,+, and therefore has composition factors S+,Sm+1, Sm+2,. . . , S-. Using (1.1), (5.1), (5.2), and (5.3) we get [2m] - = 2dimF Im ',

so that dirn P Im/Im,1 f. Thus, the p-rank of 7 ,1 is the same as the rank in characteristic zero. We have proved Proposition 5.4.3. coker r/, has no p-torsion.

Proof of Theorem 1.2.1.

Proposition 4.2.1 proves part 1.2.1.a.1. while Propositions 1.2.2 and Proposition 5.4.3 prove part 1.2.1.b. Finally, part 1.2.1.a.2 is Propositions 5.4.1 and 5.4.2.
LI

We conclude with a related result.

Theorem 5.4.4. The invariant factors of the adjacency matrix A for the graph F are p2m-l-i with multiplicity di for 0 <, i < 2m - 1. Proof If f - p, then Lemma 2.3.1 shows that all eigenvalues of A are nonzero modulo t. Thus, A is invertible over a field of characteristic f, and so there can be no ï¿½ torsion in coker A. Next, view A as a matrix over the p-adic integers, and consider 772m-1,1 as an endomorphism of ZC1 by identifying x' E AC2m-1 with x E ï¿½i. We then have A = J - 12,-1.

As noted at the beginning of this section, we have the splitting ZC = ZPl ï¿½ Ker J.

But

A(1) = p2m-11

62

and A = 72m-1,1 on Ker J. The result then follows from Theorem 3.6.3.
11

REFERENCES

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the 0(5, q) Generalized Quadrangle For Odd q, Geometriae Dedicata 39 (1991),
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4 (1995). 295-316.
[6] A. E. Brouwer. W. H. Haeiners. H. A. Wilbrink. Some 2-ranks, Discrete Mathittattes 106/107 (1992), 83-92.
[7] D. de Caen, A Note on the Ranks of Set-Inclusion Matrices, (submitted), 2001. [8] D. de Caen, E. Moorhouse, The p-rank of the Sp(4, p) Generalized Quadrangle.
(preprint), 1998.
[9] C. W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, New York, 1962.
[10] P. Dembowski, Finite Geometries, New York, 1968. [11] W. Feit, The Representation Theory of Finite Groups, North Holland, Amsterdam, 1982.
[12] A. Frumkin, A. Yakir, Rank of Inclusion Matrices and Modular Representation
Theory, Israel Journal of Mathematics 71 (1990), 309-320.
[13] C. D. Godsil, Tools From Linear Algebra, Handbook of Combinatorics, MIT,
(1995), 1705-1748.
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(odes. IUEF T'rans. njforin. 1h(oi-y 14 (1968). 182-188.
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1233-1237.
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Difference-Set Cyclic Codes, Bell System Tech J. 45 (1966), 1057-1070.

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Geometries, J. Sci. Hiroshima Univ. Ser. A-I 32 (1968), 381-396.
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[24] M. Klemm, Uber die Reduktion von Permutation Moduln, Math. Z. 143 (1975),
113-117.
[25] E. S. Lander, Symmetric Designs: An Algebraic Approach, Cambridge, 1983. [26] J. M. Lataille, P. Sin, P. H. Tiep, The Modulo 2 Structure of Rank 3 Permutation
Modules for Odd Characteristic Symplectic Groups, (preprint), 2001.
[27] M. W. Liebeck, Permutation Modules for Rank 3 Symplectic and Orthogonal
Groups, Journal of Algebra 92 (1985), 9-15.
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J. Alg. Discr. Meth. 2 (1981), 333-340.
[29] F. J. MacWilliams, H. B. Mann, On the p-rank of the Design Matrix of a Difference Set, Inform. and Control 12 (1968), 474-489. [30] J. C. Massey, Threshold Decoding, Cambridge, 1963. [31] P. McClurg, On the Rank of Certain Incidence Matrices Over GF(2), European
J. Combin. 20 (1999), 421-427.
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[33] M. Newman, Integral Matrices, New York, 1972. [34] J. Rushanan, Combinatorial Applications of the Smith Normal Form, Proceedings of the Twentieth Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Fl., 1989) 73 (1990), 249-254.
[35] N. S. N. Sastry, P. Sin, The Code of the Regular Generalized Quadrangle of
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[36] P. Sin. The Elementary Divisors of the Incidence Matrices of Points and Linear
Subspaces in P"(Fp), Journal of Algebra 232 (2000), 76-86.
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its Standard Module, To appear, J. Algebra, 2001.
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in P'(Fq), (preprint), 2000.
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Statist. Mimeo. Series 561 Chapel Hill, 1967.
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Block Designs. II, Some PBIB Designs Based on Symplectic Geometry over
Finite Fields, Acta Mathematica Sinica 3 (1965), 362-371.
[41] R. M. Wilson, A Diagonal Form of the Incidence Matrices of t-Subsets vs. kSubsets, European J. Combin. 11 (1990), 609-615.
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of the General Affine Group, J. Combin. Theory &cr. A 63 (1993), 301-317.

BIOGRAPHICAL SKETCH

I was born to Robert Duane and Nancy Claire Lataille in Bedford, New Hampshire, on December 19, 1973. I lived with my parents and older brother Robert Duane II in the town of Manchester, where I attended Green Acres Elementary School. In 1985, the Lataille family moved to Middletown, New Jersey, where I attended River Plaza Elementary School and Thompson Middle School. In 1987, the family relocated to Clearwater, Florida. There I attended Tarpon Springs Middle School and East Lake High School. After high school, I attended the University of Florida, where I earned Bachelor of Arts and Master of Science degrees in mathematics, and also played the drums in professional music groups with Devin and Brendan Moore, Aaron Carr, and Jared Flamm. But most importantly, while at the University of Florida, I was fortunate enough to meet my future wife, Angela Cuevas.

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Peter 7n , Chairman
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Pham Huu lp
Associate P sor of Mathematics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, scope and quality, as a dissertation for the degree of Doctor of Philosoph

Hemtl Vokeklein
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

iMeera Sitharam
Associate Professor of Computer and Information Science and Engineering

This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.

December 2001

Full Text

PAGE 1

THE ELEMENTARY DIVISORS OE INCIDENCE MATRICES BETWEEN CERTAIN SUBSPACES OE A EINITE SYMPLECTIC SPACE By JEFEREWklCHAEL LATAILLE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OE THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2001

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ACKNOWLEDGEMENTS I would like to thank my advisor Dr. Peter Sin, to whom 1 am grateful for his help and guidance. 1 would also like to thank Dr. Chat Ho, Dr. Pham Huu Tiep, Dr. Helmut Voelklein, and Dr. Meera Sitharam for attending several of my talks and for offering helpful comments and criticism. n

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TABLE OF CONTENTS ACKNOWLEDGEMENTS ii ABSTRACT v CHAPTERS 1 INTRODUCTION I IT Description of the Main Problem I 1.2 Statement of the Main Result 3 1.3 Background on Incidence Problems 4 1.4 Outline of the Solution 10 2 REPRESENTATIONS OF G IN ODD CROSS CHARACTERISTIC . 12 2.1 Preliminaries 12 2.2 The Complete Submodule Lattice 14 2.3 The Dimensions 15 3 REPRESENTATIONS OF G IN CHARACTERISTIC 2 20 3.1 Preliminaries 20 3.2 Restricted Action 20 3.3 The Composition Factors of Res^^ 23 3.4 The /cGAf-Composition Factors of k'^'^ 26 3.5 The fcG-Composition Factors of 30 3.6 The /cG-Submodule Lattice of 36 4 DETERMINATION OF THE p'-TORSION 41 4.1 Preliminaries 41 4.2 Determination of the p/-torsion in coker 17^,1 45 5 DETERMINATION OF THE p-TORSION 48 5.1 Preliminaries 48 5.2 Related Structure Theorems 49 5.3 The Modules Mr 51 5.4 Determination of the p-torsion in coker rjr,i 58 REFERENCES 63 iii

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-.-. ; ^'.'7 .\.M Â• Â• Â“ V ~ v ofiyp tr fj^ 7y raj?" Â• ' ' Â• ,ji^.^'-.V'.,-. 1Â“ Â•Â•. Cr . .. BIOGRAPHICAL SKETCH 66

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE ELEMENTARY DIVISORS OF INCIDENCE MATRICES BETWEEN CERTAIN SUBSPACES OF A FINITE SYMPLECTIC SPACE By Jeffrey Michael Lataille December 2001 Chairman: Dr. Peter Sin Major Department; Mathematics Let p be an odd prime and m ^ 2 an integer. Let V be a 2m-dimensional Fp-vector space equipped with a non-singular alternating bilinear form ( , ). For 1 ^ r ^ 777 let denote the set of r-dimensional isotropic subspaces of V' and let C 2 m-r denote the set of orthogonal complements of r-dimensional isotropic subspaces of V. Between any two sets Cr and Cs we have an incidence relation given by inclusion of subspaces. This information can be encoded in an incidence matrix, a 0-1 matrix which can be read in any commutative ring. Thus, it is natural to ask for the elementary divisors of this matrix as an integer matrix. In this dissertation, the integral invariants are determined for the cases in which one of the sets is Â£i.

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CHAPTER 1 INTRODUCTION 1.1 Description of the Main Problem Let p be an odd prime and let U be a 2m-dimensional Fp-vector space equipped with a non-singular alternating bilinear form ( , ). We say that the form induces a symplectic geometry on V and we refer to V together with (, ) as a symplectic space. If 777. = 1 then V is called an hyperbolic plane. We exclude this case from consideration and shall always assume that m ^ 2. We say that the vectors r. ir G V are orthogonal U ( r. w) = 0. Given a subspa.ee R C V we define R'^ := {u G U I {v,w) = 0 for all w G R}, and call R^ the orthogonal complement of R. A subspace R C V is called isotropic if R C It is a standard fact from linear geometry (e.g., see Artin [1]) that V has an isotropic subspace of dimension r if and only if 0 ^ r ^ m. For 0 ^ r ^ m, let denote the set of r-dimensional isotropic subspaces of V and let C 2 m-r denote the set of orthogonal complements of r-dimensional isotropic subspaces of V . There is a natural incidence relation between any two of the sets Cr and Cs given by inclusion of subspaces. We can encode this information in an incidence matrix as follows: Fix integers r.s such that 0 ^ .s ^ r ^ 2m. and assume that we have some fixed but arbitrary orderings on the elements of Cr and Cs-, so that we may write and Cr Â— {Ri , . . . , R|Cr|} -Cs Â— {Ri, Â• , 5'|Â£^|}. 1

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2 Define a |Â£s| x |Â£r| matrix rjr,s = by _ f 1 if C Rj Â“ \0 if 5, ^ Rj We call rjr^s an incidence matrix between Cr and Â£5. It is easy to see that if ry' ^ is another incidence matrix between Â£^ and Â£j (i.e., with respect to some other ordering on the sets Â£^ and Â£5), then r/'^g = Prjr^sQ^ where P and Q are permutation matrices of appropriate sizes. Since all of the entries in are either 0 or 1, we may regard T]r^s as a matrix over any commutative ring. We will be interested in regarding ry^^s as an integer matrix. We pause now to recall some basic facts from matrix theory (e.g., see Newmann [33]). Two integral matrices A and B (of the same size) are called equivalent if there exist invertible integral matrices C and D such that B = CAD. Each integral matrix A is equivalent to a matrix of the form diag{di , . . . , dr, 0, . . . , 0), called the Smith Normal Form (SNF) of A. Here each d, is a non-zero integer, d{ divides d,q.i for all 1 ^ ^ r Â— 1, and r is the rank of A. The elements {d,} are unique up to associates and are called the invariant factors of A. The prime-powered factors of the invariant factors, counted according to their multiplicities, are called the elementary divisors of ,4. knowledge of the invariant factors is equivalent to a knowledge of the elementary divisors. Regarding r]r,s as an integral matrix, it is natural to ask for its elementary divisors. We remarked above that all the incidence matrices between Â£^ and Cs are equivalent, and therefore it follows that these elementary divisors are independent of the chosen ordering on the sets Â£^ and Â£j. Thus, for our purposes we may speak of the incidence matrix between Cr and CgIn the cases in which one of the sets is Â£0, the SNF is easily computed. In fact, for 0 < r < 2m, the SNF of rjrfi is the row matrix (1 0 0 ... 0 )

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3 of size 1 X |Â£r|. We now consider the two extreme cases in which one of the sets is C\. It is easy to see that the SNF of 772 m, i is the column matrix (l\ 0 0 W of size |Â£i| X 1, while the SNF of 7714 is the identity matrix of order |Â£i|. In this dissertation, we shall be concerned with the intermediate cases in which one of the sets is C\ \ i.e., our problem is to determine the elementary divisors of r]r,\ for all 2 ^ r ^ Â‘2m Â— 1. 1.2 Statement of the Main Result Denote by Z^Â’' the free abelian group on the set CrThe incidence matrix between C\ and Cr can be interpreted as the Z-module homomorphism 77^,1 : Z^" -) Z^Â‘ which sends an isotropic r-space to the (formal) sum of the 1-spaces it contains. Then finding the elementary divisors of the incidence matrix is equivalent to finding a cyclic decomposition of the cokernel of this homomorphism. /p-i Let di be the coefficient of -Din the expansion of I f I . Note that di \j=o / is equal to the number of monic monomials in 2m variables which have total degree equal to i{p Â— 1) and are such that the degree in each variable is at most p Â— 1, (c-g-, see Hamada [18]). Then 2m Â— 1 d, = d 2 m-i and ^ d, = i=l 2m 1 1 . ( 1 . 1 ) j p In (1.1) we are using the />-binomial coefficients n~ LS J p n 1=1 pn-,+1 _ I pÂ® Â— 1

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4 This is the number of s-dimensional subspaces in an n-dimensional vector space over Fp. We now state our main result. Theorem 1.2.1. Assume 2 ^ r ^ 2m Â— 1. (a) If r / m, then the incidence matrix between Cr and C\ defines a finite abelian group with cyclic factors of the following orders: (1) with multiplicity 1; (2) with multiplicity cf for 1 ^ i ^ r Â— 1 (b) The incidence matrix between Cm o-nd C\ defines a free abelian group with free p(p+ l)(pÂ™-i 1) rank equal to 2(p1) Remark 1.2.2. Replace p by any prime power q (even or odd) in the general setup above, so that R is a non-singular symplectic space over the finite field F,. Our methods can be used to establish the following results: (a) For r ^ m, the only p'-torsion in coker is a cyclic factor of order [!(]^ . (b) We have rankiF rjrn.i Ki q{q^ + 1 ) 2 ( 9 1 ) for any field F whose characteristic is different from p. For odd q, these assertions can be established simply by replacing p by g in our arguments. For even q, the situation actually simplifies for the following reason: the most difficult part of the p'-torsion in the coker for us to compute is the 2-torsion; when q is even, 2-torsion is no longer p'-torsion and we are in fact able to use our methods for handling the odd p'-torsion to handle all of the p'-torsion in this case. 1.3 Background on Incidence Problems The main problem which we solve in this dissertation (see Theorem 1.2.1) is part of a wider class of problems on incidence. To put our problem in the proper

PAGE 10

5 perspective, we introduce the general notion of an incidence structure and define its associated incidence matrix. We then discuss how such a structure may be useful and indicate ways by which it may be studied through its incidence matrix. Finally, we present several examples which are closely related to the topic of this dissertation. A finite incidence structure is an ordered triple S = {V,B,X) which consists of two disjoint finite sets V and B., and a subset I of the cartesian product V x B. The elements of V are called points and the elements of B are called blocks. If the ordered pair (p, B) is in X, then we say that p is incident with B. Put \V\ = V and \B\ = 6, and let the points and blocks be labeled as follows: V = {pi,...,py} and B= {B,....,Bk}. An incidence matrix for the incidence structure <5 is a u x 6 matrix .4 = (ctij) such that Let F be any field. We denote by Cf the column null space of A over F and call it the code over F of the incidence structure. We denote by Cp the column space of A over F and call it the orthogonal code over F of the incidence structure. If F has characteristic p, then the dimension of Cp is called the p-rank of A. In general, the only incidence structures which are either mathematically interesting or practically applicable are those which possess some degree of regularity. One of the broadest classes of such regular incidence structures is the class of tactical configurations (e.g., see Dembowski [10]). These are incidence structures with equally many points on every block and equally many blocks through every point. Many configurations of classical geometry are of this kind (e.g., the configurations of Desargues and Pappus in projective geometry, and those of Miquel and the bundle

PAGE 11

6 theorem in inversive geometry). A tactical configuration with equally many blocks through every two distinct points is called a balanced incomplete block design (BIBD), or design, for short (e.g., see Assmus et ah [2]). These structures are studied not only by combinatorists and coding theorists, but also by statisticians, who use them for the planning and analyzing of agricultural and other experiments. It is often the case that the most accessible way to study an incidence structure S is through its associated incidence matrix A. For example, an analysis of the three matrices A, AA\ and A^A permits valuable insight into the combinatorial structure of S (e.g., see Dembowski [10], pg. 19-21). Thus, we may sometimes reduce a combinatorial or coding theoretic problem to one about integral matrices. The Smith Normal Form (SNF) has been used extensively in the study of integral matrices (e.g., see Newmann [33]), and so it is not surprising that it has proven effective in studying incidence matrices. We now discuss several ways in which the SNF of an incidence matrix may be of use. Consider first the common situation of deciding when two combinatorial structures are isomorphic. Often we may view these combinatorial objects as incidence structures, and may therefore consider their corresponding incidence matrices. An isomorphism between the two objects translates to an equivalence of the two incidence matrices by permutation matrices. Since permutation matrices are unimodular (i.e., invertible over the integers), we then see that if the two objects are isomorphic then their associated incidence matrices have the same SNF. We remark that the converse is not true; i.e., nonisomorphic objects may have the same SNF. See Rushanan [34] for an example where two strongly regular graphs associated to latin squares are shown to be nonisomorphic by comparing the invariant factors of their associated incidence matrices. Another application of the SNF is in computing p-ranks of incidence matrices. Indeed, the p-rank of an incidence matrix A is equal to the number of invariant

PAGE 12

7 factors of A which are not divisible by p. These p-ranks are important to coding theorists. In fact, let F be a finite held of characteristic p, S a BIBD, A the incidence matrix associated to S, and C\ the corresponding code. Then CV has merit in that it can be decoded by a relatively simple procedure known as majority logic decoding (see Massey [30]). In order for Cf to serve effectively as an errorcorrecting code (see Assmus et al. [2]), it is necessary for the dimension of Cw to be large. This is equivalent to requiring the p-rank of A to be small. For an example of the SNF being used to study p-ranks of incidence matrices associated to a class of designs called symmetric designs (i.e., designs in which the number of points is equal to the number of blocks), see Lander [25]. We now discuss several important classes of incidence structures and the progress which has been made on the study of their associated incidence matrices. Let n be a natural number and put N := {1, . . . ,n}. For each 1 ^ A; ^ n let Tk be the set of all subsets of N of cardinality k. We consider the incidence structure St,k whose points are the elements of Tk, whose blocks are the elements of Tt, and where incidence is given by set-theoretic inclusion. The associated incidence matrix will be denoted by At,kWe note that St,k is a tactical configuration, and that, when A: = 1, it is actually a design. It has been known for quite some time that At,k has full rank over a field of characteristic zero (see Gottlieb [15]). The 2-rank of At,k was given by Linial and Rothschild [28]. In the same paper, the authors also determined the 3-rank in the special case where f = A: -|1. The answer for any prime is given by Wilson [41]. In fact, he determines a diagonal form for At^k which is equivalent to the SNF. Recently, de Caen [7] has derived a recurrence relation for the rank (over most fields) of At^kNow let p be any prime number and q = pk We denote by V an n -f 1dimensional vector space over F, . Then (resp. T'^) will denote the set of rdimensional linear (resp. affine) subspaces of V. We consider the incidence structure

PAGE 13

8 Ss,r (resp. (S'^) whose points are the elements of J-r (resp. whose blocks are the elements of J-g (resp. and where incidence is given by inclusion of subspaces. We denote by As^r (resp. 5s, r) the associated incidence matrix. We note that
PAGE 14

9 Having looked at several related examples, we now restate the main problem of this dissertation in the language developed in this section. Let g = p* be a power of the odd prime p and let V be a non-singular symplectic space of dimension 2m over F, . Then, for 1 ^ r ^ m . Cr will denote the set of isotropic subspaces of VÂ’ and ^ 2 m-r will denote the set of orthogonal complements of isotropic subspaces of V. We are interested in the incidence structure whose points are the elements of Â£i, whose blocks are the elements of Cr, and where incidence is given by inclusion of subspaces. We let pr,i denote the associated incidence matrix. Note that this incidence structure is a tactical configuration which is not a design; this follows from the observation that two 1-spaces are contained in a common isotropic subspace precisely when they are orthogonal to one another. For more information on this structure, see Wan ZheXian [40]. In the special case when m = 2 and t Â— 1, de Caen and Moorhouse [8] worked out the p-rank of p 2 ,iThis result was then generalized by Sin [.37], who found the p-rank of ?p,i for 1 ^ r ^ 2m Â— 1 and t = 1. In this dissertation, we first derive those elementary divisors of i]r,i (1 ^ ^ ^ 2^ ~ 1) which are coprime to p; this result is established for all t. Thus, as a corollary we may deduce the f-ranks of these matrices for all primes i ^ p. In the special case when t = 1, we are also able to determine the p-elementary divisors of pr,! (1 ^ ^ 2m Â— 1). We remark that for r ^ m the elementary divisors asserted in Theorem 1.2.1 are precisely the same as those found in [36]. This fact can be interpreted in coding theoretic terms as follows: if r / m, then the code coming from the design of 1-spaces vs. arbitrary r-spaces of V is the same as the code coming from the tactical configuration of 1-spaces vs. isotropic r-spaces of V. To actually implement these codes in practice, it is necessary to generate the associated incidence matrices on the computer. Since the number of isotropic r-spaces in V is smaller than the number of arbitrary r-spaces in V', we see that it is computationally more effiecient to implement the (equivalent) code coming from the isotropic subspaces.

PAGE 15

10 1.4 Outline of the Solution Let G := Sp(2m, p) be the symplectic group for V. Then G is the set of all linear transformations from V into itself which leave the form invariant. It follows from WittÂ’s Theorem (e.g., see Artin [1]) that G acts transitively on each of the sets Cr, for 1 ^ r ^ 2m Â— 1. By extending this action linearly becomes a ZG-permutation module and the incidence map T]r,i becomes a ZGmodule homomorphism. Let I be any prime. Then is a ZG-submodule of Z^'' and the quotient Z^'Â‘/Â£Z^"' ~ is an F^G-module. Since we have an induced F^G-module homomorphism We call fjr,i the reduction mod I of r]r,i and refer to the rank of 77^,1 as the l-rank of 77r,i. Note that the Grank of 77^,1 is equal to the number of invariant factors of 77^,1 which are not divisible by 1. To prove Theorem 1.2.1 we will need to conduct a detailed study of these reduced incidence maps. In order to carry out this study, we will need to know the submodule structure of the permutation module for G on the 1-spaces of its natural module in all characteristics f. For ^ not equal to 2 or p, the complete submodule lattice has been determined by Liebeck [27]. We collect these results in the hrst section of Chapter 2 and then use elementary linear algebra to determine the dimensions of each of the submodules found in Liebeck [27]. In Chapter 3 we derive the modulo 2 submodule structure. The arguments are taken from Lataille et al. [17]. This puts us in a position to completely determine the jo'-torsion in coker 77^,1 (see Chapter 4). Although our interest is in the representation theory of G = Sp(2777,p), all of the arguments in Chapters 2, 3, and 4

PAGE 16

11 work with p replaced by q, where q is a power of p. Therefore, in those chapters only we let V denote a non-singular symplectic space of dimension 2m over F, , and we let G denote the symplectic group Sp{2rn,q). In Chapter 5 we handle the natural characteristic case. To simplify the situation, we localize at p and examine closely the permutation modules and incidence maps over Zp, Qp, and Fp. Here Zp and Qp denote the p-adic integers and the padic numbers, respectively. After first gathering structure theorems from Sin [37], we introduce certain Zp-forms Mr in QpG-modules which are isomorphic (whenever r 7 ^ m) to the kernel of the augmentation map on Qph The submodule structure of the mod p reductions Mr := MrjpMr of these lattices is the essential ingredient in the determination of the p-torsion in coker pr,i-

PAGE 17

CHAPTER 2 REPRESENTATIONS OE G IN ODD CROSS CHARACTERISTIC 2.1 Preliminaries Let 9 be a power of the odd prime p. Throughout this chapter, V will denote a non-singular symplectic space of dimension 2m over F, , and G will denote the symplectic group Sp{'2m,q). In this chapter we collect results from Liebeck [27] which give the complete submodule lattice of , where F is a field of characteristic not equal to 2 or p. We then use elementary linear algebra to compute the dimensions of each of the submodules found in Liebeck [27]. We begin by establishing some notation and presenting some definitions. It is an easy consequence of WittÂ’s Theorem (e.g., see Artin [1]) that the group G acts transitively with rank 3 on Â£i; i.e. there are three orbits for the action of G on pairs of 1-spaces of V. Equivalently, we may say that the stabilizer in G of each 1-space has three orbits on Ci. For x Â€ Â£i let {x}, A(x), and $(x) be the orbits of Gx on El, where Gx denotes the stabilizer in G of x. If we take A(x) = {y e El \ y ^ x-^}. then |A(x)l = q^-i and and |$(x)| 1 ) q-1 |A(x)nA(2/)] = Q^-^(q-l) 12

PAGE 18

13 fov X ^ y in Â£i with y G A(a;).. Denote by F the strongly regular graph on Â£i associated to G in which x is joined to y if and only if y G A(a;). Recall that a graph is called strongly regular if all vertices have the same valency and if the number of vertices adjacent to both of two distinct vertices ui and V 2 depends only on whether v\ and t >2 are adjacent or not. We define the adjacency matrix (call it .4) of F as follows: Fix some arbitrary ordering on the elements of Â£ 1 , so that we may write Â£1 Then A = (ai,j) is the |Â£i| x |Â£i| matrix such that _ f 1 if XjG A(xj) |0 if Xi ^ A(xj) Â’ Now let F be any field and let F^* be the associated permutation module for FG. For a subset B C Ci, write SB beB In particular, we write 1 := x^Ci There is a natural, non-singular, FG-invariant symmetric bilinear form on F^' defined by demanding Â£1 be an orthonormal basis, and then extending the action linearly to the whole space. For any subset W C F^* we write W-^ := {n G F^' I [v,w] = 0,Vte G IT} and call W'^ the orthogonal complement of W in F^Â‘ . Note that we have used the same terminology and notation for orthogonal complements in V, but no confusion should arise since it will always be clear from context which type of complement is intended. We remark also that if W is an FG-submodule of F^Â‘ , then so is W-^.

PAGE 19

14 2.2 The Complete Submodule Lattice Define FG-submodules UÂ±qm-\ of as follows: UÂ±qm-i := + 5a(x)) [Â±q"'~^y + s^f^y)) | G Â£i) , where ( ) denotes F-span. Following [27] we call UÂ±qm-\ the graph submodules of F^Â‘ . We will require the following nontrivial result due to Liebeck [27]: Lemma 2.2.1. (cf. Liebeck [27], Theorem 1). // F is a field of characteristic I, where Â£ p, then any WG -submodule ofW^'which is not contained in (!) contains a graph submodule. Using this result, Liebeck is able to determine the complete submodule lattice of the permutation module for G on the 1-spaces of its natural module in all characteristics other than 2 and p. Lemma 2.2.2. (cf. Liebeck [27], pg. 10). Assume F has characteristic i, where Â£ = 0 or else char i ^ p is odd. (a) If Â£ = 0 or if Â£ > 0 but Â£ \ [^[Â”]^, then the graph submodules are nonisomorphic, simple â€¢G-modules and we have the orthogonal decomposition F^* = (l) 0 UÂ—q-m-l 0 Uqm-\. (b) If Â£ > 0 and if Â£ \ q'^ I, then Uqm-i is a simple â€¢G-module, but D (1) . The quotient U_qm-ij ( 1 ) is simple and not isomorphic to Uqm-\. Furthermore , we have F"^Â‘ = U-Â®Uqm-., where Uis a uniserial â€¢G-module which contains as a submodule of codimension one. We have

PAGE 20

15 and = (l) 0 Ugm-1. (c) If i > 0 and if i \ [7],Â» then UqmÂ—\ is a simple FG-module, but Ugm-i D (1) Â• The quotient Ugm-xj [\) is simple and not isomorphic to U_gm-i. Furthermore , we have = LG Â® U-gm-l , where [/+ is a uniserial WG-module which contains Ugm-i as a submodule of codimension one. We have (U_grr.-.)^ = U+ and (Ugm-l)'^ = (1) 0 U_gm-1 . The structural information given in Lemma 2.2.2 will be used heavily in Chapter 4. For certain results we will also need to know the exact dimensions of each of the submodules of . We compute these dimensions in the next section. 2.3 The Dimensions We now compute the dimensions of the submodules of F^Â‘ for char F ^ 2,p. Our strategy is to view the adjacency matrix A for the graph F as an FG-module endomorphism of F^Â‘ (in the natural way) and to recognize the direct summands of F^' as the generalized eigenspaces for A. We begin with a result from Higman [20]: Lemma 2.3.1. (cf. Higman [20]). // F is an arbitrary field, then the characteristic polynomial of A over F is c(x) = (xq^^-^)(x + q"^-^y(x yghere f = and g = 2 ( 9 1 ) Next we will need the following computational results:

PAGE 21

16 Lemma 2.3.2. If x 6 Ci, then A{Â±q^-^x + 1)1 + Â±^"'(Â±9"Â“'^ + ^A(x))Proof. For x ^ C\ we have A[Â±q^~'^x + SA(r)) = + A(sa(i)) = ^ ^A{y)yeA(x) But an easy computation shows 5A(y) = 9^"^ ^(1-a;), J/Â€A(r) and the result follows. Lemma 2.3.3. fFe have A^ -q^^-'^I = q^^~\q-\)J, where J denotes the all-one matrix. Proof. If a; Â€ Â£i, then A (x) = ^(>SA(a:)) j/eA(i) and so it follows from the proof of Lemma 2.3.2 that A^ = q^^-^ J q^^-\j I) With these facts in hand we may compute the dimensions of the FLi-submodules of F^Â‘ whenever the characteristic, of F is not equal to 2 or p : Proposition 2.3.4. If i ^ 2 ov p, then dimf U-gm-i = / and dimj Uqm-i = g.

PAGE 22

17 Proof. We proceed by cases. Throughout the following, let K\ denote the generalized eigenspace for A corresponding to the eigenvalue A. Note that since A is an FGmodule homomorphism, each of its generalized eigenspaces is an FG-submodule of F^T CASE 1; Â£ = 0. In this case A is diagonalizable (.4 is symmetric) and from Lemma 2.3.1 we see that A has minimal polynomial m{x) = {xFrom Lemma 2.3.2 we deduce PÂ—qmÂ—l C and Uqm-l C A q-mÂ—l . It then follows from Lemma 2. 2. 2. a that these containments are actually equalities. Thus, dimy U_qm-\ = /, and dirriy Uqm-i = g. For the remaining cases, assume that f > 0 is odd and unequal to p. CASE 2: Â£\q^"^-l. The minimal polynomial, m(x), for A over F must divide {x q^^-^){x^ Since are all distinct modulo i, it follows that m{x) = {xq^--^){x^ q^--% and that A is diagonalizable over F. We see from Lemma 2. 2. 2. a and Lemma 2.3.2 that A^gmÂ— 1 = IJ^qTnÂ—l.

PAGE 23

18 As in Case 1, we have dim^Y = / and dimf Uqm-i Â— g. CASE 3: i\q-\hntl \ Since and are distinct modulo Â£, the polynomial Â— divides m{x). But by Lemma 2.3.3 we know that m(x) divides . We must then have m(x) = x^ Â— and hence A is diagonalizable over F. Since mod we see from Lemma 2. 2. 2. a and Lemma 2.3.2 that Â— UÂ„qTnÂ—l and I\qm-l Â— (l) 0 h qm-\ . Thus, the conclusion follows in this case. CASE 4: Â£ I g 1 and Â£ I As in Case 3, A is diagonalizable over F with minimal polynomial m(x) = X^ g 2 m2 _ Since I is odd by assumption and since gcd{qÂ— 1, gÂ™ + 1) = 2, we must have Â£ \ Â• Since g^Â™~^ = gÂ™~^ mod it follows from Lemma 2.2.2.C and Lemma 2.3.2 that AÂ—qmÂ—l Â— U^qn and I\qjnÂ—l Â— So diniY LAqm-\ = / and diniY Uqm-\ = g. CASE 5: ^|g 1 but ^ I The minimal polynomial of A must divide (x-g2Â™-^)(x^-g^Â™-2).

PAGE 24

19 Since ^ and Â—q^ ^ are distinct modulo we see that m{x) is divisible by 2 2m-2 X Â— q . Since Â£ \ q Â— 1, Lemma 2.3.3 shows that m{x) = (x ~ Since gcd{q"' + 1, [^Jg) = 1 or 2? we must have i \ q'^ + I or Â£ \ . If the former holds, then q^Â™'~^ = Â—q^~^ mod and so Lemma 2.2.2.L and Lemma 2.3.2 show that Â— f/_ and I\qmÂ—l Â— U qmÂ—l . Since q'^~^ is not repeated as a root of m{x), we see that diruf Uqm-i = g, the multiplicity of as a root of the characteristic polynomial of A. We then deduce dirriY U= 1 A f. On the other hand, if Â£ \ ? then = q^~^ mod and then Lemma 2.2.2.C and Lemma 2.3.2 show I\ Â—qmÂ—l Â— U ^qin~l and I\qm Â— l Â— U+Since Â— gÂ” ^ has multiplicity one as a root of m(x), we see that dirrif = /, and hence dim^Uj^ = I A g.

PAGE 25

CHAPTER 3 REPRESENTATIONS OE G IN CHARACTERISTIC 2 3.1 Preliminaries Let Q be a power of the odd prime p. Throughout this chapter V will denote a non-singular symplectic space of dimension 2m over F, , and G will denote the group Sp(2m,g). In this chapter we determine the complete FGÂ’-submodule lattice of F^Â‘ , where F is a field of characteristic 2. The arguments are taken from Lataille et al. [26]. Our approach is to first restrict the action of G to that of a certain maximal subgroup, namely the stabilizer in G of a maximal isotropic subspace of V. The composition factors of this restricted action are determined and using a recent result from Guralnick et al. [17], we are then able to determine the composition factors for the action of the full group. This puts us in a position to obtain the submodule lattice (see Theorem 3.6.3). For simplicity, we will always work over an algebraically closed field of characteristic 2, which we denote by k. 3.2 Restricted Action Fix a symplectic basis ei , . . . , , /i , . . . , /Â„ for V over F, , so that = {fÂ„fj) = 0 and (eÂ„ /_,) = -(/j, e Let M := (ci, . . . , em) and P := (/i, . . . , /Â„) be maximal isotropic subspaces of V. Let Gm denote the set-wise stabilizer of M in G. Then Gm = S>^L (3.1) 1 if i = j 0 if i ^ j 20

PAGE 26

21 where ^={(o f) M = (3.2) and i=|(* j-,) |9Â€GL(M)|. (3.3) Here / is the m x m identity matrix and 0 is the m x m zero matrix. To determine the /cG'-composition factors of we will first need to determine the composition factors of Res^^ , where Res^^ denotes the /cGM-inodule obtained by restricting the action of G on to that of GmWe start by noting that Gm has two orbits on C\ : C>i := {u; 6 Â£i I o) C M] and O2 Â€ C-\ I uj ^ M } . Now for any subset X C C\, we will let denote the fc-span of the elements of Then we have the following decomposition of Res^^ as a direct sum of kG A/-submodules: Resg^ = k^' Â© k^G (3.4) Thus, to determine the composition factors of Resg^ we may separately study the summands in (3.4). The first summand is easily handled: For n Â€ K write v = ( ^ ) , where a; Â€ M and y ^ P. Then \yj (v) G O 2 if and only if y ^ 0. (3.5) With this notation, the computation / /a;\ _ /x T Ay\ 0 /J[yJ-[ y ) (3.6)

PAGE 27

22 shows that S acts trivially on Oi, i.e. S acts trivially on From (3.1), (3.2), and (3.3) we see that the induced action of Gm jS ~ GL(A/) on is the usual action of GL(M) on the 1-spaces of M. Thus, the /cGw-submodule lattice of k'^' is known from Klemm [24]. Explicitly, if we put and lo, := ^ u; e k^^ weOi K. := {u> Â— a I where ( denotes A;-span, then we have Lemma 3.2.1. (a) If m is odd, then K, is simple and (b) If m. is even, then k'^^ is uniserial with composition series k^^ fC { 0 } In the situation of Lemma 3.2.1.b, put 1C := ICj {Ioi)k Â• We will indicate the composition factors of informally by writing i.Oi _ j k + Ki if m. is odd , ~ |(2)A: -b A.'' if m is even ' Of course, here k denotes the simple trivial module. To determine the /cG'M-composition factors of the second summand in (3.4), we will once again begin by restricting the action of Gm to that of its normal subgroup S; i.e., first we will determine the composition factors of Resf" k^^ . We will then

PAGE 28

23 use CliffordÂ’s theorem (e.g., see Curtis et al. [9] or Feit [11]) along with Lemma 2.1.2 and Proposition 2.2.4 to recover the GM-composition factors. 3.3 The Composition Factors of Res^^ Using elementary linear algebra we see that given any non-zero y ^ P and any z ^ M we can always find a symmetric transformation A 6 Hom(P, M) which sends y to z. Therefore, it follows from (3.6) that the P-orbits on O 2 are indexed by the 1-spaces in P. Explicitly, let (t/i ),..., \ be a list of the 1-spaces in P. Then the 5'-orbits on O 2 are the sets 0<Â„, -{((;,)) I XÂ€M,}. Thus, we have the following decomposition of Res^^ as a direct sum of /cP-submodules: Â‘-1 q-l Resf^ = 0 kÂ°(y1=1 Let Sy, ^ S he the stabilizer of .9,, I .4 0 / Â€ 0(y,). So e .9 I Ay, = 0 ^ . Then k^M = Ind^ k, so that by (3.8) we may write (3.8) 9-1 Resf"^ kÂ°^ = 0 Ind|^_fc. (3.9) 1=1 Recall (e.g., see Curtis et al. [9] or Feit [11]) that for an arbitrary group G, a subgroup PI ^ G, and a kH-modu\e M, we denote by Ind^M the fcG-module induced from the kH-modu\e M. Explicitly, we have Ind^M = kG ^kH Al.

PAGE 29

24 We now pause to establish a correspondence between the irreducible kScharacters and the symmetric bilinear forms on M. This correspondence will be the key to determining the composition factors of Res^^ . We start by noting that P ~ V/M = V/M^ ~ M*, (where M* denotes the dual space of M) so we may identify P with M* . If we also identify M with (M*)*, then we can identify Hom(TÂ’, M) with Hom(M*, (M*)*); i.e., we may regard Hom(R, M) as the set of all bilinear forms on M*. The correspondence (o (3.10) then identifies S with the set of symmetric bilinear forms on M*. Under this identification Sy^ corresponds to the set of all symmetric bilinear forms on M* which have y, in their radical; i.e., Sy^ corresponds to the symmetric bilinear forms on {Ke?' yi)* . Now let be a primitive p-th root of unity in algebraically closed k. The corresondence /(.) (3.11) allows us to identify the linear functionals on the F, -vector space S with the irreducible ^-characters of the elementary abelian p-group S. Since S is the set of symmetric bilinear forms on M*, we see that S* is the set of symmetric bilinear forms on M. Thus, we may identify the irreducible characters of S with the symmetric bilinear forms on M. Remark 3.3.1. Let N be an irreducible submodule of Ind|^ k and let / G 5* be the linear functional which corresponds under (3.11) to the character of N. By Frobenius reciprocity (e.g,, see Curtis et ah [9] or Feit [11]), we know that Sy^ acts trivially on N. This means that Tracefjp^{f{A)) = 0 for every A G , from which it follows

PAGE 30

25 that f{A) Â— 0 for all A ^ Sy^. But as Sy, is the set of symmetric bilinear forms on {Ker yi)*, this means that the symmetric bilinear form on M which corresponds to / must be isotropic on the hyperplane Ker yi C M. Thus, the irreducible characters in Ind|^ fc are the symmetric bilinear forms on M which are isotropic on Ker y{. Again using Frobenius reciprocity, we see that each such form occurs with multiplicity one. In particular, the zero form (which corresponds to the trivial character) occurs exactly once in each Ind^^ k. In fact, it is easily seen that the unique trivial submodule of Iud|^ k is where lÂ°(Â„) E Â“ ^ (3.13) Now let B he a non-zero symmetric bilinear form on M which has an isotropic hyperplane. Then B has either rank 1 or 2. If B has rank 1, then the radical of B, denoted by Rad B, is the unique isotropic hyperplane for B. li B has rank 2 then M/Rad B is hyperbolic and therefore has precisely two isotropic lines for the form induced from B, i.e. M has precisely two isotropic hyperplanes for B. For all of this linear geometry, see .4rtin [1]. In light of (3.9), the above then gives us all of the composition factors of Res^" k'^'^. We record this information as Lemma 3.3.2. Under the identification in (3.11), Res^" k^'^ has the following composition factors: (a) The zero form, i.e. the trivial character, which occurs with multiplicity . (b) The rank 1 symmetric bilinear forms, where each occurs with multiplicity 1. (c) The rank 2 symmetric bilinear forms having isotropic hyperplane, where each occurs with multiplicty 2.

PAGE 31

26 3.4 The fcG\f-Composition Factors of We start by examining the 5Â’-fixecl points of k'^'^ . Define T ;= 0 7; (3.14) t=i where the Ti are as in (3.12). Now it is easily seen from (3.1) that GmIS Â— GL(M) permutes the vectors in (3.13) in the usual way that GL(M) acts on the 1-spaces of M*; i.e., in the usual way that GL(M) acts on the hyperplanes of M. Thus, if we let Cm-i denote the set of hyperplanes in M, then the A;GA/-module T can be naturally identified with the fcGL(M)-module It is well-known (e.g., see Dembowski [10]) that the permutation modules on the 1-spaces and the hyperplanes, respectively, of M are isomorphic over a field of characteristic zero. Therefore, from a general principle of modular representation theory (see Feit [11], Theorem 17.7) we know that and have the same composition factors. Therefore, it follows from (3.7) that rp _ j k + 1C ifmis odd , ^ Â“ {{2)k + lC if m is even ^ ^ We remark here that it can actually be shown that k^'^~' and k^'are isomorphic for G. To find the remaining composition factors, we now consider the action of Gm on the irreducible characters of S. We start by observing that as S acts trivially on its characters, we need only consider the induced action of GmIS ~ GL(M). Now GL(A/) acts by congruence transformations on S. Therefore, if we view the elements of S* as symmetric matrices, then the action of GL(M) is again by congruence transformations. We then see that under correspondence (3.11), GL(M) acts by congruence transformations on the characters of 5'.

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27 There are two GL(M) congruence classes of rank 1 symmetric bilinear forms, represented by cliag{l,0 , . . . , 0) (3.16) and diag{a, 0, . . . , 0) (3.17) where a is a non-square in (see Artin [1]). The stabilizer of both classes is Â±1 * which has index in GL(Ty). Let Bi denote the congruence class of (3.16) and Ba denote the congruence class of (3.17). Let Wi denote the external direct sum of the S'-characters which correspond to the forms in Bi, and let W 2 denote the external direct sum of the Scharacters which correspond to the forms in BaThen it follows from Lemma 3.3.2.b and CliffordÂ’s Theorem that has composition factors, call them Wi and W 2 , which when restricted to S are isomorphic to Wi and W 2 , respectively. Note that dimk Wi = dimk W2 = (3.18) but Wi ^ W 2 . Now, there is one congruence class of rank 2 symmetric bilinear forms having isotropic hyperplane, represented by iVo) 0 0 0 (3.19) The stabilizer in GL(M) of this class is where C is a 2 x 2 monomial matrix. This subgroup has index ^ ^ GL(M).

PAGE 33

28 Let D denote the external direct sum of the 5-characters which correspond to these forms. Note that dirnk D = follows from Lemma 3.3.2.C and CliffordÂ’s Theorem that exactly one of the following cases holds for k'^'^ : Case A: has precisely two composition factors, call them T>i and P_i, which when restricted to 5 are isomorphic to D. Case B: k'^^ has a single composition factor, call it Vq, which when restricted to 5 is isomorphic to D Â© D. We now show that the former is true. We start by establishing some notation which we will use throughout the remainder of the paper: For any field F, we let F^'^ denote, as usual, the FG-permutation module on Cr, and we let fjr,s : F^^ ^ F^* be the FG'-module homomorphism which sends each isotropic r-space to the (formal) sum of the isotropic s-spaces which are incident with it. We now require some notation which we introduced in Â§2.1. For the sake of exposition we repeat the dehnitions here. Define 1 := ^ X Â€ F^Â‘ iGCi For each x G Â£i put A(x) := {y G Â£i I y ^ x-^}, and define an element s^(x) ^ by Â•Sa(x) ^ X(3-20) ye^(x) Define a non-singular symmetric bilinear form ]f by demanding that the elements of Cl form an orthonormal basis. For any subset 5 C F^Â‘ put 5'-*:= |u G I [u,s]f = 0, for all s G 5}

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29 Note that we have used the same notation for orthogonal complements in V, but no confusion should arise. Note also that if S' is a FG-submodule of , then so is S~^. Now let Q 2 denote the field of 2-adic numbers and let Q 2 be its algebraic closure. Then T will be the maximal unramified extension of Q 2 in Q 2 (see Iwasawa [22], pg.37), and 77. will be the valuation ring of T . Note that T has residue field k. From Lemma 2. 1.2. a and Proposition 2.2.4 we have that = (1)^ Â© U-x Â© Ux, (3.21) where UÂ±i are irreducible iFC-submodules with dimy^ U-x = 2(7-1) (3.22) and dimjr Ux q{q^ + l){q m Â— 1 1 ) 2(7-1) (3.23) Let UÂ±i be the reductions modulo 2 of UÂ±i. Their restrictions to Gm must be collections of the composition factors described above. By (3.22) and (3.23) the dimensions of the composition factors of UÂ±x add up to ^ 2{q-f ) -Assume now that {m,q) / (2,3). Then q{q^Â±l){q"^-^Tl) 2(7-1) < 2{dimk D). (3.24) So it cannot be that either UÂ±x contains a composition factor which when restricted to S is isomorphic to D Â® D. Thus, we deduce that Case .A. holds. If {m,q) = (2,3), then dimjr U_x = 2[dimk D). However, it is easy to see (e.g., by considering degrees) that U-x is the unique non-trivial composition factor which is common to both and Since = IndS,, it then follows from Frobenius reciprocity that Gm (and hence S) has a non-zero fixed point on U-xThis then implies that Res^^ U-x contains a trivial composition factor. Since S has no fixed

PAGE 35

30 points on D Â© D, we deduce that Case A holds in this case as well. We mention here that it will be shown in Â§3.5 (see (3.32)) that T>x ~ X>_i. Hence, we have found all the /cG'M-composition factors of . Combining this information with Lemma 3.2.1 and using the informal notation of (3.7) and (3.15), we may now state Lemma 3.4.1. (a) If m is odd, then Resg^ = [2)k + {2)IC + >Vi + W Â’2 + Vx + P-i(b) If m is even, then Resg^ = (4) A: + {2)}C + Wi + W 2 + A + P-i3.5 The A:G-Composition Factors of k^'^ Let T,TZ,UÂ±i, and UÂ±i be as in Â§2.3. It follows from (3.24) and the remarks following it that each of VÂ±i occurs in exactly one of Resg^ UÂ±\, and that the VÂ±i do not occur together. Thus, we may assume that our notation is chosen so that VÂ±i is a composition factor of Resg^ UÂ±i. Also, since dimk U\ Â— dim;, T>i = q{q^ + l){q^-^ 1 ) 2{q-l) 1 9-1 1 2 q{q^ l){q"^-^ 1 ) 2(9-1) = dimk VVi = dimk W2, we deduce upon inspection of Lemma 3.4.1 that ^1 f fC + Vx {k + IC + Vx [m odd) (m even) (3.25) Suppose for the sake of contradiction that Ux has a A:G'-composition factor, call it AÂ’, which when restricted to Gm is isomorphic to /C. Since S acts trivially on

PAGE 36

31 1C, it is contained in the kernel, call it J, of the representation of G on K. Since S is not contained in the center of G, and since the center of G is the only non-trivial normal subgroup of G, we deduce that J must be all of G. But Gm acts non-trivially on /C, a contradiction. It follows that U\ has no such composition factor for G, and therefore U\ is irreducible if m is odd. If m is even, then similar reasoning allows us to conclude that either Ui is irreducible or else Ui = k + X, with X irreducible. We now show that the latter is true. Using the notation in (3.20), we define a A:G'-module homomorphism ^ ^ (3.26) by where cu G Â£i. Put a; !->Â• u; + Sa(u..)5 Now define U := Im (f. (3.27) U' : Â— [ui U2 I Ui,ii2 C U)i^ Â• (3.28) In the terminology of 2.1 we see that U' is the (unique) graph submodule over k. It is easily seen that L\ n -77.^ is an '7^-form of tq and a pure '7?.G'-submodule of . Therefore, is a mod 2 reduction of Ui as well as a fcG-submodule of k^'^ . Since Ui fl 7?.^' is certainly not contained in ( 1 )^, we see from Lemma 2.1.1 that U' CUiH 7^^l

PAGE 37

32 and therefore U\ contains the composition factors of U'. We require the following result: Lemma 3.5.1. If m is even, then (1)^^, C U'. Proof. Let M Â€ LmThen an easy computation shows that + Â»A(uj) = 1(3.29) ujCM Ul^C\ Now the number of 1-spaces in M is ? which is an even number since m is even. Thus we may group the summands in the left-hand side of (3.29) into pairs. The result then follows from the definition (3.28) of U' . Since U\ fl 7^^' has at most 2 composition factors, it follows from Lemma 3.5.1 that = u'. We may summarize the above as Lemma 3.5.2. and U' have the same composition factors. (a) If m is odd, then U' is simple. (b) If m is even, then U' is uniserial with composition series U' I ( 1 ). { 0 } In the situation of Lemma 3.5.2.b, we put U' / {1)^. := X. In the situation of Lemma 3. 5. 2. a, we will use X' to denote the composition factor isomorphic to U' . In view of Lemma 3.5.2, we have only to determine the composition factors of U-\. We do this now:

PAGE 38

33 A simple matrix computation shows that = 0. (3.30) Thus, U C Ker cp, (3.31) where U is as in (3.27). Since (f is symmetric, we see that Ker
PAGE 39

34 as was promised in Â§3.4. It remains to determine the A:G-composition factors of jU. By inspecting Lemma 3.4.1, we see that Resg^ U^IU = Wi + WÂ’2. We now show that has A;(TT-composition factors, call them fVi and 14' 2 . which when restricted to Gm isomorphic to WÂ’l and WV We will need to consider the conformal symplectic group For brevity, we put F := CSp(2m,
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35 multiplication by l3~^ on the characters of S. Taking j3 = a, where a is as in (3.17), it is now easy to see that the conjugate by g of the form in (3.17) is the form in (3.16). The result now follows from the construction of the W[s in Â§2.3. Thus, U^IU is simple for F, and it follows from CliffordÂ’s theorem that U-^/U is semi-simple for G. Now, either U'^ /U is a simple fcG-module, or else jU ~ FFi Â© IT 2 , where Wi and W 2 are simple /uG-modules which when restricted to Gm are isomorphic to Wi and W 2 , respectively. Consider the following result from Guralnick et al. [17]: Lemma 3.5.5. (cf. Guralnick et al. [17], Theorem 2.1) Any irreducible kG-module of dimension less than ^ Â— 2 (g+ i ) Â— ^ either the trivial module, or a module of dimension Â— .^n irreducible A;G'-module of dimension will be called a Weil module (see Guralnick et al. [17]). This result shows that we must have jU ~ W\ Â© VF 2 , where W\ and W 2 are irreducible Weil modules of dimension Hence, we now have all of the kGcomposition factors of . If we let Vj and V 2 be submodules such that U C Vj, V 2 C U'^ and Vi/U ~ FFi and V 2 /U ~ W 2 , then the above arguments yield the following filtration of : kÂ‘^'/\ V, V 2 (3.33) U { 0 }

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36 3.6 The fc(j-Submodule Lattice of By the minimality of U' (see Lemma 2.1.1) it suffices to determine the submodule structure of {U')^ jU' . We start by defining submodules C and C'^ as follows; C := Im fim,\ and CÂ’+ ;= (x + y 1 .r,y Â€ C)^. We will need the following results: Lemma 3.6.1. (a) C"*" C C (b) EomkG{k^-,W,) = {0}, for z = 1,2. (c) C has no quotient isomorphic to Wi, for z = 1,2. (d) Eomkoik^"' , k) ~ k (e) is the unique maximal submodule of C. Proof. An easy computation shows that \x Sa(x)) 1, for all X e Cl and all M G LmFrom the definition of we then deduce that C'-^ C (C'*')'*-. Now (a) follows by taking orthogonal complements. We have Resf W\ ~ W,, for i = 1,2. But from our construction of the W, in Â§2.3, we know that they are fixed point free for S. Therefore, Gm has no fixed points on the Res^^ Wi ~ Wj-; i.e., HomfcGM(^:ResG^ W,) = {0}, for z = 1, 2. Since k, the assertions in (b) follow from Frobenius reciprocity. Since C is a homomorphic image of we see that (c) is an immediate consequence of (b).

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37 Again by Frobenius reciprocity, we have HomA,GM(^b^) This proves part (d). It follows from (a) and (d) that C'*' is the unique maximal submodule of C with trivial quotient. From Lemma 2.1.1 we have U' C C. Using the inner product computation at the start of the proof, we have C C Thus, U' CC C {UYSince {U')^/{U')^k + k + Wr + W2, we know that any maximal submodule of C with non-trivial quotient must have quotient HÂ’l or IU 2 , which is impossible by (c). Then (e) follows. Since is not orthogonal to C, we get C n c c, and hence CnC^ c c+. by Lemma 3.6.1. e. Thus, the quotient C/{Cr\C~^) has at least 2 composition factors. Furthermore, C/{C fl C-^) has a unique maximal submodule, namely C'^ j{C Fl C-^). Lemma 3.6.2. (a) C/{C Fl U-*-) is self-dual. (b) C/(C F C'^) has a unique maximal submodule and a unique simple submodule. Both the head and socle of Cf{C F C'^) are trivial. Proof. The form induced by [Â— , Â—]k on the quotient C f{C F CÂ‘Â‘Â‘) is non-singular and therefore induces an isomorphism between C /{CClC-^) and its dual. Since the form is

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38 G-invariant, this is actually a A:G-isomorphism, and (a) follows. Part (b) then follows immediately from the remarks following Lemma 3.6.1. In light of Lemma 3.5.4, it follows from CliffordÂ’s theorem that any AjP-module having at least one of the W{ as a composition factor for G must have the other as well. Since G and G''" are modules for L. we deduce from Lemma 3.6.2 that either C/(CnC^) = k + k (3.34) or Cy(CnC^) = k + k + Wi + W2. (3.35) Suppose by way of contradiction that (3.34) holds. By Lemma 3.6.2.b it must then be the case that G/(G Cl G-*-) is uniserial. But as G is perfect, it has no module which is a non-split extension of the simple trivial module by itself. So (3.35) holds and it follows that G = (G')''' and C~^ = We may now state our main result: Theorem 3.6.3. Using the above notation, has the jollowmg submodule lattice:

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39 m even m odd kÂ“~' X C C+ Wi / \ VKi Wi \ {C^Y k 1 ^ ( 1 ) I { 0 } c ( 1 )c+ Vl Vo (C+)^ X' / \ fc ;i) c { 0 } Proof. Let be a /cG-submodule of Assume N ^ {0} or (1). Then we know from Lemma 2.1.1 that U' C N. If we assume that N ^ k^'or (I)"*", then we have that 7^ {0} or (1). But then from Lemma 2.1.1 we have U' C N-^', i.e., N C Thus, U' CN C [U'Y. From the remarks immediately following Lemma 3.6.2, we know that U' Â— and . Thus, \i N ^ U' or {U'Y , it follows from Lemma 3.6.2.L that V C N C V rl But as jU ~ VFi 0 Vf'T and since Vf', ^ ITT we see that Vi and V are the only fcG-submodules between U and U-^\ i.e., A^ = Vi or VVI

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40 Although the dimensions of the submodules pictured above have been given earlier, for convenience we recall here that + 1 ) dinik C = 1 + 2(9-1) and dimk Vi = dirrik V 2 = 2m _ I 2(9-1) Rtinuik J.6.4From Curalnick el al. [17] we know ihaL the Weil uiudules can be realized over F2 if and only if q = Â±1 mod 8. If g = Â±3 mod 8. Then the smallest held of dehnition for the W'eil modules is F4. W'ith this insight, we may deduce from Theorem 3.6.3 the complete FG-submodule lattice of for any held F of characteristic 2. Explicitly, if 9 = Â±1 mod 8 and F is arbitrary, or if q = Â±3 mod 8 and F4 C F, then the submodule lattice of F^Â‘ is as pictured in Theorem 3.6.3. However, if q = Â±3 mod 8 and F4 ^ F, then the submodule lattice is as pictured in Theorem 3.6.3 except that the quotient C '^ is irreducible.

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CHAPTER 4 DETERMINATION OE THE p'-TORSlON 4.1 Preliminaries As in the preceding two chapters, we let qhe a power of the odd prime p, and C be a non-singular symplectic space of dimension 2m over F, . We denote by G the associated symplectic group Sp(2m,g). For any field F, we let : F^" -> F^' denote, as usual, the incidence map between F^Â’' and F^^ . Also, we let : F^' -) F be the FG'-module homomorphism which sends each /? G to 1 C F. We call G the augmentation map of F^"" . In the case where r = 1 we shall simply write e instead of Si. Note that Cr may be identified with the incidence map fir.o : F^'' Â— >Â• F^Â” . We present two lemmas, the proofs of which are easy computations: Lemma 4.1.1. Assume 2 ^ r ^ m. If x G Gi and R G Cr, then ( if X C R + < 0 ifxCR^,x<^R [ qÂ’Â’~^ if X ^ R^ Lemma 4.1.2. Assume 1 ^ r ^ m. If x G Ci and R-^ G C 2 m-r, then ( if -r C R [V2,r,-rA{R^]. Â±q^~' T + S J 1 + Â‘ v/ T C R\ X % R \ g2m-r-l ^ ^ These simple computations yield the following useful result. Proposition 4.1.3. Assume W is a field oj characteristic i ^ p41

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42 (a) For r ^ m we have 77^,1 (Ker Sr) / UÂ±gm-i . (b) If ^ 2 we have Imip fjm,i / U^gm-i but Imp f]m,i -L Ugm-i. (c) If Â£ = 2, then U^gm-i = Ugm-i and Imp fjm,i -L Ugm-\. Proof. Let a:i, X 2 , . . . , j/i, j/ 2 , Â• Â• Â• , yÂ™ be a symplectic basis for V, so that {xi,xj) = {y^,yj) = 0 and (xi,j/j) = Assume that 1 ^ r ^ m Â— 1 and put Ri := (xi, . . . x^) and R 2 := {x 2 , Â• , Xr+i) Then Ri, R 2 ^ L^. If 2 ^ r ^ m Â— 1, then using Lemmas 4.1.1 and 4.1.2 we see that %ARi) Vr.i{R2), iÂ±q"-' {X2) + 5 a((x,))) (iyÂ™"' (yi) + ^a((.,Â»)] = -y'"' + o, and \r\rAR\) fir.liRt). (Â±yÂ™"' (^2) + ^A((.,Â») (Â±y"^Â“' (yi) + ^a((.,)))] = ^ OÂ’ which establishes the assertion in part (a) for these cases. If r = 1, then [yr,i(/?^) fjrAR^), (Â±r-' (y 2 ) + ^A((,,))) " (Â±yÂ™"' (yi) + ^a((Â„)))] = VÂ”'"' ^ o. proving the assertion in part (a) for this case as long a.s Â£ ^ 2. Suppose Â£ = 2. Then there is a unique graph submodule, call it IR, and by Lemma 4.1.2 we see that [ipARf], ((xi) + ^A((X,))) ((Â‘^Â’ 2 ) + Â•Â«A((x2Â»)] = 1/0. This shows that Im 772 m1,1 / 17i. (4.1) Since is odd, we see that r 1 if i = j |o if i / j Im 772 m-i,i / Ker e. (4.2)

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43 Then equations (4.1) and (4.2) imply that Im ? 72 m-i,i = Â• (4Â’3) To see this, suppose for the sake of contradiction that Im r) 2 m-i,i ^ Â• Then by (3) we deduce that (Im is a non-zero submodule of not contained in (1) . So by Lemma 2.2.1 it must contain U\, contradicting (2). But then (4) implies ? 72 m_i,i(Ker Â£r) Â— Ker e and part (a) is then seen to be true in this case as well. To prove parts (b) and (c), let M Â€ Cm-, so that M = M-^. We see from Lemma 4.1.1 that [VmAM), (q^-^x + SA(.)) + 5a(,))] = 0 for all x,y ^ Cl, but ['/m,l(T/), (Â— ^X + Â— (Â—q'^ ^y + SA{y))] = if X C M and y ^ M. Parts (b) and (c) now follow. We now deduce an important characterization of ^r,i(I'ler Sr)Proposition 4.1.4. If F is a field of characteristic Â£ p and if r m, then 77r,i(Ker e^) = Ker e. Proof. We know that ?7r,i(Ker Er) C Ker e. Suppose this containment were strict. Then we would have (Ker e)-*C (?T,i(Ker Cr))"^But (Ker e)-^ = (1). so Lemma 2,2.1 then says that (r),.,,(Ker Â£^))-^ contains a graph submodule. By Proposition 4.1.3 this is impossible.

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44 We finish this section with the following: Proposition 4.1.5. Let F be a field of characteristic ^ ^ 2,p. (a) If i = 0 or if i > 0 but I \ , then Imp i?m,l = (1)Â® U-qm-i. (b) If I > 0 Â£ I gÂ™ + 1, then Imp fm.x = U-(c) If f > 0 and f \ [7]^ Â• Imp /7m, 1 = (1)Â® ILqm-l. Proof. We begin by noting that Imp 77 m, i is a nontrivial FG-submodule of F^Â‘ which is contained in by Proposition 4.1.3. Suppose first that the hypotheses in (a) hold. Since Imp 77m , 1 ^ Ker e, the assertion in (a) follows immediately from Lemma 2. 2. 2. a. Under the hypotheses in (b), we see from Lemma 2.2.2.b that either Imp '/7m,i = Lor Imp 77m , 1 = . But since I\ , we see that Imp 77m , 1 ^ Ker e, and the result follows.

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45 Finally, if ^ | [7], > then Lemma 2.2.2.C shows that either Imp hm,l = (1) 0 U-qm-l or Inir Â— U-qni-i. Now given M G Cm we see that A[f,mAM)) = q^-\l nraAM)), showing that Imp T]m,i ^ K-gm-1 UÂ—qm Â— l (see Cases 4 and 5 in the proof of Proposition 2.3.4), so that we must have Imp r], m,l ( 1 ) 0 U-qn 4.2 Determination of the /j/-torsion in coker Vr i Let T]r,i : Z^Â‘ and ^ denote the incidence and augmentation maps, respectively, over Z. When r = 1 we shall simply write e instead of 5i. We may now prove Proposition 4.2.1. If r m, then the only p' -torsion in coker r]r,i is a cyclic factor of order [[]^. Proof. We have e(Z^Â‘) = Z and e(lm Z,

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46 and so 'LÂ‘~' l[\m l]r,l + Â£) Â— Z. ( 4 . 4 ) Since r ^ m we see from Lemma 2 . 2 . 2 . a and Proposition 4 . 1.3 that r)r,i is surjective over a field of characteristic zero. Thus, Im^ rjr^i has full rank, and so coker 77^,1 is a finite group. Therefore, from ( 4 . 4 ) we deduce |coker 77^,1! = |(Ini 77^,1 + Ker e)/Im 77 r,i| x So we must show that Ker e/(Ker e fl Im 77^,1) is a jo-group. Let be a prime different from p. Since ( 4 . 5 ) 77^,1 (Ker Cr) = Ker e fl Im 77^,1, we need to show that where 77 r,i(Ker Sr) = Ker e. Ker Er = Ker Cr/iKer Er and 77 r,i is the reduction mod I of 77^,1Since Ker Er is a pure submodule of Z^K we may identify Ker Er = Ker Â£r/Â£Ker Er = Ker Â£r/(I'^er Er H with the image of Ker Er in Z^''/fZ^Â’' = F^'', which is (Ker Er + Â«^Â’")/^Z^Â’' = Ker Â£,, where Er denotes the reduction mod f of Â£r. Thus, we need to show that 77^,1 (Ker Er) = Ker Â£. But this last fact has been shown in Proposition 4 . 1 . 4 .

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47 Taking q = p \xi Proposition 4.2.1, we see that Theorem 1.2.1.a. 1 has been established. We conclude this section with the following: Proposition 4.2.2. Assumt f ^ p hs primt. Thtn coker i]rn,i has no i-torsion. Proof. We must show that dinifi Im f]m,i = rankz Im for all primes Â£ / p. If is odd, this follows from Proposition 4.1.5 along with Proposition 2.3.4. If Â£ = 2 then this follows from Theorem 3.6.3.

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CHAPTER 5 DETERMINATION OE THE p-TORSION 5.1 Preliminaries Throughout this chapter V will be a non-singular symplectic space of dimension 2m over Fp. We denote by G the associated symplectic group Sp(2m,p). In light of Propositions 4.2.1 and 4.2.2 we are have only to determine the p-elementary divisors in order to prove Therem 1.2.1. Thus, we shall simplify the situation by localizing at p. Let Zp denote the ring of p-adic integers and Qp the field of p-adic numbers. Let p,,3 : Zj^ ^ Zj* and Â£r ' ZpÂ’Â’ Â— ^ Zp denote the incidence and augmentation maps, respectively, over Zp, and let Pr,s : Fj^ ^ Fj^ and Â£r If'p'^ IFp denote their reductions mod p. Put y,. := Ker Â£, . For a ZpG-module A we shall use the notations QpA := Qp Zp A and A := AjpA. Since p I |Â£r I, we have the splitting zj^ = Zpi Â© y; 48

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49 and hence also = Fpl Â© ?r. It is easily seen that ^r,s(^r) ^ SI and consequently ^]r,s ( 1 r ) ^ 1 s Â• Since the p-torsion for rj,.,s comes from its action on V, . we will restrict onr attention to these submodules, and shall use the same notation for the restricted maps. We will sometimes need to consider the incidence maps i]r,s between all the r-spaces of V and all the s-subspaces of V. 5.2 Related Structure Theorems We begin by collecting several results from Sin [37]. The first result gives the submodule structure of V'l.

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50 Lemma 5.2.1. (cf. Sin [37], Theorem 1). The WpG-rnodule Vi has the following submodule lattice: Vi ITi IT2 Wm /\ VT+ IT_ \/ hhm+l iV 2771Â—1 { 0 } Fori ^ m, the quotients S{ := Wi/Wi+i are simple. The quotient VTm/hLm+i is the direct sum of two non-isomorphic simple modules, call them and S-. is chosen so that W.^IWm+\ Â— -5'+ o-fid W-jWm.\\ Â— 5'_. Our notation From Sin [37] we also have that dzmpp Si = di for i ^ m, (5.1) dimpp S+ + dimw,, S= dm. (5.2) diiuf^ S+ diirif^ S= pÂ™, (5.3) and Si ~ S 2 m-i as FGÂ— modules. (5.4)

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51 Now Cl and Â£ 2 m-i are isomorphic as G-sets, via the map which sends each 1space to its orthogonal complement. Thus, the permutation modules Y\ and Y 2 m-i are isomorphic, and it follows from Lemma 5.2.1 that Y^m-i has the following structure: h 2 m-l f 2m Â— 1 f 2m Â— 2 Cm /\ C+ [L \/ Um-l 1 L Ui { 0 } Here we have f/j/fh-i ~ Si for i Y and UmICm-i Â— 0 6 _. Our notation is chosen so that U+jUm-i Â— 5Â’+ and U-IUm-i Â— 5'_. We need in addition the following fact. Lemma 5 . 2 . 2 . (cf. Sin [37], Lemma 4). W has a unique maximal submodule with simple quotient isomorphic to Sr z/r 7 ^ m, and S^ if r = m. 5.3 The Modules Mr Let Mr := W/Ker 7 /^, 1 We have Mi = Yi and M 2 m-i = Y 2 m-\Since Ker 77^,1 is pure, Mr is a Zp-form in the QpG-module QpW/QpKer 77,., 1 . When r / tu, we have QpV;/QpKer 77r,i ~ QpV'i.

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52 It follows from a general principle of modular representation theory (e.g., see Feit [11], Theorem 17.7) that in this case all of the mod p reductions Mr have the same composition factors as VÂ’l and V 2 m-iThe following simple observation will be used frequently. Let m. Then it is easy to check that r Â— t s Â— t Vr,t It then follows that 7]r,s(Ker rjr,i) Q Ker Thus, we have induced ZpG-homomorphisms (5.5) 1.1 r : Mr M, and along with the relations ^ /^r,s Pr,s Mr Â— > Mg r Â— t s Â— t Pr,t and f^s,i Â® /^r,s J p r Â— t s Â— t J p Lemma 5.3.1. For 2 ^ r ^ 2m Â— 1 we have Ker rjrp = Ker ? 7 r, 2 m1(5.6) Proof. We know from Sin [36] that Ker r)r,i = Ker Pr, 2 m-iSince W C V7, and since r]r,i and r]r,2m-i are just the restrictions of fjr,i and i)r:2m-\ to Vj.. we have Ker T]rp = Ker rjrp Pi Vj= Ker i)r;2m-i P V7 Â— A e? I]r.2m Â— 1Â‘ From Lemma 5.3.1 it follows that we have induced maps Pr,2m~l ' Mr Â—> '^2m-X

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53 and iUr,2m-l : Lemma 5.3.2. For we have the relation r]s,r Â° T]t,s = [^l|] Vt,rProof. This follows by taking transposes in (5.5). Lemma 5.3.3. For I ^ s ^ r ^ m Â— 1. we have r]s^r{F^^' Vs.i) Q hr. Proof. We know from Sin [36] that ^r,2mÂ— 1 ^ Vs,r Â— so that 2m Â— 1 Â— s r Â— s ^?s,2m Â— 1 ? fis,r{F-er ?/s,2m-i) ^ Ker 7%, 2 m Â— 1 Â• Restricting 7)5, r, Vs,2m-i, and rjr,2m-i to Vj, and W, respectively, gives 775 ,r(Ker r]s, 2 m-i) Q Ker 77^, 2 mThe result then follows from Lemma 5.3.1. From Lemma 5.3.3 we see that there are induced maps l^s.r Â• T/r and Ps,r ' Ms Mr, for 1 ^ s ^ r ^ 7?r Â— 1. The next result gives the submodule structure of MrLemma 5.3.4. Assume 2 r ^ m Â— 1 (a) Mr has a unique maximal submodule with simple quotient isomorphic to Sr-

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54 (b) The maximal submodule of Mr is the direct sum of a uniserial module Jr = Ker fr,!, which maps isomorphically under flr, 2 m-i to Ur-i, and a uniserial module Jf = Ker fr, 2 m-\, which maps isomorphically under jlr,i to Wr+\. Proof. As Mr is a homomorphic image of Yr, the property in part (a) follows from Lemma 5.2.2. To prove (b), we observe that by part (a) the image of the non-zero homomorphism p,r,i in Vi must be either W,. or fV 2 m-rBut the dimension ol Im frp is equal to the p-rank of pr.i, which by Sin [37] is equal to dim^j, BA > dim lT 2 m-rSo Im pLr,\ = BA. Thus, the kernel Jf has composition factors S\,. . Sr-iSimilarly, Im pr, 2 m-\ equals Ur or U 2 m-rBut since Ker pr,i = Ker we see that the dimension of the image of Pr, 2 m-\ is equal to the p-rank of Pr, 2 m-i) which is equal to the p-rank of p 2 m-r,iIt is shown in Sin [37] that this common rank is equal to dim^p Ur < dimfp fAm-rSo Im Pr, 2 m-i = UrThus, the kernel Jf has composition factors ?m-l, 5'_, We claim that Jf n Jf = {0}. In fact, if r ^ m 1, let N be the smallest submodule of Mr with two composition factors isomorphic to 5V+i Â— S 2 m-r-ii and if r = m Â— 1, then take N to be the smallest submodule of Mr with composition factors isomorphic to both 5'+ and S-. Then the above remarks show N C J + , so that dimfpN ^ dimfpJf. But by definition of N, we must have pr,i {N) = BA+iSince dimfpWr+i = dimYpJf, we then have dimf^N ^ dimf^Jf. It follows that N = Jf and that pr,i is injective on N, i.e. Jf fl Jf = {0}. Hence, each map is injective when restricted to the kernel of the other. So Jf is isomorphic to a submodule of Ur, which from inspection of composition factors must be Ur-iAlso, as shown above we have Jf ~ ITr+iLemma 5.3.5. For 2 ^ r ^ in Â— 1, we have Ker fr,r-i Â— Jf and Im fr,r-i Â— J^-i,

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55 Proof. Suppose for the sake of contradiction that Im jlr,r-i % Then using Lemma 5.2.2 we see that (Im Pr,r-\) is either Ur or U 2 m-rBut from Lemma 5.3.4 we know that Im fir-\, 2 m-i = Ur-iSince t/^-i S \$ U 2 m-r-, we have obtained a contradiction. So Im pr,r-i C J^i. Again using Lemma 5.2.2 we then see that the image of /Ur,r-i is either proper submodule of which has {5'2m-r, Â• Â• Â• , 5Â’2m-i} as its set of Composition factors. By (5.5), we have /^r,l ~ A^2,1 O * Â‘ * 0 fr Â— 2.r Â— \ ^ /^r,r Â— 1' (5.7) up to a non-zero scalar in Fp. It follows that the rank of fr,\ is not more than the rank of fir,r-iSince, by Lemma 5.3.4, the former map has rank equal to since ~ we must have Im fr,r-\ = JY\But (5.7) also shows Ker fir,r-\ C TÂ”. Since diruf^ Ker fir,r-i d,, this containment must actually be equality, and the assertion is proved. Define to be the number of r-dimensional isotropic subspaces of a non-singular 2m-dimensional symplectic vector space over Fp. Explicity, 2m r r Â— 1 = n (pÂ”" + ') J !Â«0 i-0 m r We will need the following computation. Lemma 5.3.6. .AssumeÂ‘2 ^ r ^ 7v Â— l. Then the eigenvalues ofi]r^or]\ r are (^) ['Tliso ['""21 i.. multtplictty 1. (ii) ^ multiplicity /, (Hi) p''~^ o'*] Â• multiplicity g. \ * L r Â— 2 A iso 1 Proof. A straightforward computation shows that 2m Â— 2 r Â— 1 / + iso 2m Â— 4 r Â— 2 [J-A-I). Vr,l Â° m,r J tso

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56 Since /, J, and A are commuting, diagonalizable transformations, we may simultaneously diagonalize them. But J has rank equal to one with 1 as eigenvector. The result then follows from Lemma 2.3.1. D Corollary 5.3.7. 7/2 ^ r ^ m Â— 1, then Im Â° Mi,r = {0}Proof. This is immediate from Lemma 5.3.6. Lem ma 5.3.8. IJ 2 ^ r ^ nz 1, then Vp o^, Â— i,r)} [ ^ 1* Proof. If r = 2, the claim is seen to be true by Lemma 5.3.6. Assume that the claim is true for all numbers smaller than r, and put C := H 2 ,l O iU3,2 o Â• Â• O jXr^r-l 0 0 Â• Â• O jj. 2 , 2 , Â° /^ 1 , 2 Then we see that r Vp[det{C)) = ^{up(det(;tq_i,,)) -|Up(det(p,,,_i ))} i=2 ~ ^ ^ f . ^ 1 Â— 1 ^ T^r Â— l,r)) ,=2 Vl -Ip / Â— (r 2) I 1 I Â“h Vp Â— 1 ^ /^r Â— l,r)) Â’ J p by induction. But using (5.5) along with Lemmas 5.3.2 and 5.3.3, we see that C = 7^r,i 0 Mi,r up to a unit in Zp. Thus, by Lemma 5.3.6, we have Vp{det{C)) = (r 1) 2m 1 -Ih p and the assertion follows. Corollary 5.3.9. 7/2 ^ r ^ m Â— 1, then Im fir-\.r ^ {0}.

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57 Proof. If the linear transformation ^r-i,r were identically zero, we would have Vp{det{^r-l,r)) ^ = dimiFp Mr-1. But from Lemma 5.3.5 we know that r-l Up((iet(/ir,rÂ— 1 ) ) ^ ^ ^ i=l = dimpp J", 2 m 1 so that Vp(detillr-l.r)) + Vp{det{l.lr,r-\)) ^ im 1 r-l 1 + ^ d, J P / !=1 > 2 77? 1 1 , j p which is impossible by Lemma 5.3.8. So Im fJ,r-i,r is a non-zero submodule of MrLemma 5.3.10. For 2 ^ r ^ m Â— 1, the following hold: (a) Im p.r-l,r = Jf, (b) Ker fr-i.r = (c) Im fi^r is isomorphic to Si and is contained in J~. Proof. We proceed by induction. If r = 2, then by Corollary 5.3.9 we have Im /Ii ,2 / {0}. By Corollary 5.3.7 we have Im pi.i C J.J , and the assertion is seen to be true in this case. Assume the assertion is true for all numbers smaller than r, where r ^ 3. If Im fr-i.r ^ dÂ“, then from Lemma 5.2.2 we see that flr,i (Im fir-i,r) is either VKr-i or W^m-r+xBut from Lemma 4.6 we know that Im fr.x Â— kW. Since Wr C W^-i^ we must have fr.i (Im fr~x,r) = kkÂ’ 2 m-r+i. It follows that Ker fir,xOfir-x,r has composition factors ^i, . . . , S' 2 m-r. From the characterization of in the proof of Lemma 5.3.4 we then see that Ker fir,i Â°/Ir-i,r Â— Now from Corollary 5.3.7 we know that the

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58 composition yLr,\ o A*i,r is identically zero. But using (5.5) along with Lemmas 5.3.2 and 5.3.3, we get /^r,l ^ ^ Â— 1 up to a non-zero scalar in Fp. Thus, Im ^ /Ir.i Â°/lr-i,r = Jr-i-' contradicting the induction hypothesis. So Im /i,_i,r is a (non-zero) submodule of J~ which, by Lemma 5.3.4, has a unique maximal submodule with simple quotient isomorphic to Sr-iIt must be that Im /ir-i,r = establishing part (a). We then see that Ker has composition factors Sr, ,S 2 m-iFrom the characterization of in the proof of Lemma 5.3.4 we deduce Ker jlr-i,r = ^,^15 establishing part (b). It remains to prove that Im /ii,r ^ and that Im jli^r Â— Â•S'l. But /^l,r Â— /^r Â— l,r ^ up to a non-zero scalar in Fp, and from the induction hypothesis we get Im /ii,r-i ^ JS-i yt'i.r-i Â— Fi. Since J~-i Fl J,t_i Â— {0} and since = Ker jir-i,r, it follows that Im fi\^r K a non-zero submodule of J~ = Im Since any non-zero homomorphic image of V'l must have a unique maximal submodule with simple quotient isomorphic to S\, we see that part (c) is established. 5.4 Determination of the n-torsion in coker ? 7 r , i Proposition 5.4.1. Assume 1 ^ r ^ m Â— 1. The p-elementary divisors of the incidence matrix between Cr and C,\ are with multiplicity d, for 1 ^ z ^ r Â— 1. Proof. The image of the incidence map, between 1-spaces and all r-subspaces of V contains Imz ?/r,i, and hence coker fr,i is a homomorphic image of coker ipp-

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59 Since the asserted elementary divisors are the same as those of fjr,i (see Sin [36]), r Â— 1 and since from Sin [36] we have Up( [coker ?}r,i[) = E (r Â— i)di, it will suffice to show 1 = 1 r Â— 1 = i)diÂ»=i For 2 ^ s ^ r, Lemma 5.3.10 shows that Ker //s-i,s = ^^at 2mÂ— 1 i=s = dirrif^ and Lemma 5.3.5 shows that Ker jls,s-i Â— Jj ^ so that SÂ— 1 Cp(def(^5^3_i)) ^ ^ ^ di !=l = dinii'^ Jj . But then Lemma 5.3.8 shows that we must actually have equality in both places above, i.e. sÂ— 1 Up(def (^5^3Â— 1 ^ ^ d, t=i and 2m Â— 1 l,s)) Â— ^ ^ i=s for all s. But then (5.5) implies r Vp{det{iJ,r,i)) = Vp{det{fXs,s-i)) s=2 r sÂ—\ s=2 i = l r-I 1=1 and the result is proved. Since for x ^ Ci and R ^ Cr vje have x C R 'R and only if R^ C x-*-, we see that Proposition 5.4.1 gives the p-elementary divisors of the incidence maps ? 72 m-r, 2 m-i, for

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60 2 ^ r ^ m Â— 1. Note also that since L^m-x is precisely the set of all hyperplanes of V, the elementary divisors of are known from Sin [36]. With these observations we may prove Proposition 5.4.2. Assumt 1 < r ^ m 1. Then the p-elementary divisors of the incidence matrix between C2m-r and C\ are multiplicity di for 1 ^ i ^ 2m Â— r Â— 1. Proof It is easy to check that ^2mÂ— 1,1 ^2mÂ— r,2mÂ— 1 P ^2mÂ— r,l "f But then the induced maps satisfy r Â— 1 1 J. -> p rÂ— 1 , PÂ‘2m Â— l,l Â® /^2mÂ— r,2mÂ— 1 Â— P P2mÂ—r,l since J = 0 on V 2 m-r = Ker J. So, if we put ( [^[Â”1 p Â“ det[p2m Â— l,l)dct[p2m Â— r,2m Â— l) P^ ^ det[p2mÂ—r,l)' and so r Â— 1 2m Â— 2 2m Â— 1 Vp(det(p 2 m-r.i)) = ^ (2m. 1 0^* (r-1) ^ d, i={ i=l ^=I 2mÂ— r Â— 1 = (2m Â— r Â— i)di. i=l But the incidence map, fj 2 m-r,Xi between 1-spaces and all (2m Â— r)-subspaces of V, is such that 2mÂ— rÂ— 1 Updcoker ? 72 m-r,i|) = (2m-r-i)d,, i = l by [36]. The result then follows as in the proof of Proposition 5.4.1. From Lemma 5.2.2 we see that Impp f]m,x is a submodule of Vi which has a unique maximal submodule with simple quotient isomorphic to S+. It then follows

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61 from Lemma 5.2.1 that ImiPp ilm,i Â— + i therefore has composition factors S+,Sm+uSm+ 2 ,--.,S 2 m-i. Using (1.1), (5.1), (5.2), and (5.3) we get 2ml , Â„ T m 1 = 2d^mFp Im r]m,i P , J p so that dirrif^ Im prn,i = /Â• Thus, the p-rank of 77^,1 is the same as the rank in characteristic zero. We have proved Proposition 5.4.3. coker 77^,1 has no p-torsion. Proof of Theorem 1.2.1. Proposition 4.2.1 proves part 1.2. 1. a. 1. while Propositions 1.2.2 and Proposition 5.4.3 prove part 1.2.1.b. Finally, part 1.2.1.a.2 is Propositions 5.4.1 and 5.4.2. We conclude with a related result. Theorem 5.4.4. The invariant factors of the adjacency matrix A for the graph P are p2m-i-t multiplicity d{ for 0^7^ 2?n Â— 1. Proof Ifi ^ p, then Lemma 2.3.1 shows that all eigenvalues of A are nonzero modulo i. Thus, A is invertible over a field of characteristic Â£, and so there can be no i torsion in coker A. Next, view T as a matrix over the 79-adic integers, and consider r]2m-i,i as an endomorphism of by identifying x-^ G C 2 m-i with x G Â£1. We then have Â•4 Â— J 772mÂ— 1,1 As noted at the beginning of this section, we have the splitting ZpÂ‘ = Zpl Â© Ker J . But A(l) = p ^^-^1

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62 and A. Â— Â— ? 72 m-l,l on Ker J. The result then follows from Theorem 3.6.3.

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REFERENCES [1] E. Artin, Geometric Algebra, New York, 1957. [2] E. F. Assmus, J. D. Key, Designs and their Codes, Cambridge, 1992. [3] B. Bagchi, A. E. Brouwer, H. A. Wilbrink, Notes on Binary Codes Related to the 0{5,q) Generalized Quadrangle For Odd q, Geometriae Dedicata 39 (1991), 339-355. [4] S. C. Black, R. J. List, On Certain .Abelian Groups Associated with Finite Projective Geometries, Geometriae Dedicata 33 (1989), 13-19. [5] A. Blokhuis, Some p-ranks Related to Orthogonal Spaces, J. Algebraic Combin. 4 (1995). 29.5-316. [6] .'\. E. Brouwer. W. H. Haemers, H, .A. Wilbrink, Some 2-ranks, Discrete Mathematics 106/107 (1992), 83-92. [7] D. de Caen, A Note on the Ranks of Set-Inclusion Matrices, {submitted.), 2001. [8] D. de Caen, E. Moorhouse, The p-rank of the Sp(4,p) Generalized Quadrangle, (preprint), 1998. [9] C. W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, New York, 1962. [10] P. Dembowski, Finite Geometries, New York, 1968. [11] W. Feit, The Representation Theory of Finite Groups, North Holland, Amsterdam, 1982. [12] .A. Frumkin, A. Yakir, Rank of Inclusion Matrices and Modular Representation Theory, Israel Journal of Mathematics 71 (1990), 309-320. [13] C. D. Godsil, Tools From Linear .Algebra, Handbook of Combinatorics, MIT, (1995), 170.5-1748. [14] .). M. Goethals, P. Delsarte, On a Class of MajorityLogic Decodable CyclicCodes. IFEF Trans. Inform. I'heory 14 (1968). 182-188. [15] D. H. Gottlieb, A Class of Incidence Matrices, Proc. Amer. Math. Soc. 17 (1966), 1233-1237. [16] R. L. Graham, F. J. MacWilliams, On the Number of Information Symbols in Difference-Set Cyclic Codes, Bell System Tech J. 45 (1966), 1057-1070. 63

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64 [17] R. M. Guralnick, K. Magaard, J. Saxl, P. H. Tiep, Cross Characteristic Representations of Odd Characteristic Symplectic Croups and Unitary Croups, (submitted)^ 2001. [18] N. Hamada, The Rank of the Incidence Matrix of Points and d-flats in Finite Ceometries, J. Sci. Hiroshima Univ. Ser. A-I 32 (1968), 381-396. [19] N. Hamada, On the p-rank of the Incidence Matrix of a Balanced or Partially Balanced Incomplete Block Design and its .Application to Error Correcting Codes, Hiroshima Math. J. 3 (1973), 153-226. [20] D. C. Higman, Finite Permutation Croups of Rank 3. Math. Z. 86 (1964). 145156. [21] C. Hiss, On the Incidence Matrix of the Ree Unital, Designs, Codes and Cryptography (1997), 57-62. [22] K. Iwasawa, Local Class Field Theory, Oxford, 1986. [23] W. Kantor, On Incidence Matrices of Finite Projective and .Affine Spaces, Math Z. 124 (1972), 315-318. [24] M. Klemm, Uber die Reduktion von Permutation Moduln, Math. Z. 143 (1975), 113-117. [25] E. S. Lander, Symmetric Designs: An Algebraic Approach, Cambridge, 1983. [26] J. M. Lataille, P. Sin, P. H. Tiep, The Modulo 2 Structure of Rank 3 Permutation Modules for Odd Characteristic Symplectic Croups, (preprint), 2001. [27] M. W. Liebeck, Permutation Modules for Rank 3 Symplectic and Orthogonal Groups, Journal of Algebra 92 (1985), 9-15. [28] iN. Linial, B. Rothschild. Incidence Matrices of SubsetsÂ— a Rank Formula. SLAM J. Alg. Discr. Meth. 2 (1981), 333-340. [29] F. .]. MacWilliams, H. B. Mann, On the />rank of the Design Matrix of a Difference Set, Inform, and Control 12 (1968), 474-489. [30] J. C. Massey, Thi^eshold Decoding, Cambridge, 1963. [31] P. McClurg, On the Rank of Certain Incidence Matrices Over GF(2), European J. Combin. 20 (1999), 421-427. [32] B. Montaron, On Incidence Matrices of Finite Projective Planes, Discrete Mathematics 56 (1985), 227-237. [33] M. Newman, Integral Matrices, New York, 1972. [34] J. Rushanan, Combinatorial Applications of the Smith Normal Form, Proceedings of the Twentieth Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FI., 1989) 73 (1990), 249-254. [35] N. S. N. Sastry, P. Sin, The Code of the Regular Generalized Quadrangle of Even Order, Proceedings of Symposia in Pure Mathematics, 63 (1998), 485-496.

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65 [36] P. Sin, The Elementary Divisors of the Incidence Matrices of Points and Linear Subspaces in P"(Fp), journal oj Algebra 232 (2000), 76-86. [37] P. Sin, The Permutation Representation of Sp(2m,Fp) .Acting on the Vectors of its Standard Module, To appear, J. Algebra, 2001. [38] P. Sin, The Invariant Factors of the Incidence Matrices of Points and Hyperplanes in P"(F,), (preprint), 2000. [39] K. J. C. Smith, Majority Decodable Codes Derived From Finite Geometries, Inst. Statist. Mimeo. Series 561 Chapel Hill, 1967. [40] Wan Zhe-xian, Studies in Finite Geometries and the Construction of Incomplete Block Designs. II, Some PBIB Designs Based on Symplectic Geometry over Finite Fields, Acta Mathematica Sinica 3 (1965), 362-371. [41] R. M. Wilson, A Diagonal Form of the Incidence Matrices of f-Subsets vs. kSubsets, European J. Combin. 11 (1990), 609-615. [42] A. Yakir, Inclusion Matrix of k vs. I Affine Subspaces and a Permutation Module of the General Affine Group, J. Combin. Theory Ser. A 63 (1993), 301-317.

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BIOGRAPHICAL SKETCH I was born to Robert Duane and Nancy Claire Lataille in Bedford, New Hampshire, on December 19, 1973. I lived with my parents and older brother Robert Duane II in the town of Manchester, where I attended Green Acres Elementary School. In 1985, the Lataille family moved to Middletown, New Jersey, where I attended River Plaza Elementary School and Thompson Middle School. In 1987, the family relocated to Clearwater, Elorida. There I attended Tarpon Springs Middle School and East Lake High School. After high school, I attended the University of Elorida, where I earned Bachelor of Arts and Master of Science degrees in mathematics, and also played the drums in professional music groups with Devin and Brendan Moore, Aaron Carr, and Jared Elamm. But most importantly, while at the University of Florida, I was fortunate enough to meet my future wife, Angela Cuevas. 66

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