The number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field

Generating functions in the form of infinite products are given for the number of equivalence classes of nondegenerate sesquilinear forms of rank n over GF(q²) and for the number of equivalence (or congruence) classes of nondegenerate bilinear forms of rank n over GF(q).     

The Steinberg Lattice of a Finite Chevalley Group and its Modular Reduction

Let p be a prime and let q = p^a, where a is a positive integer.Let G 7equals; G(F_q) be a Chevalley group over F_q, with associatedsystem of roots and Weyl group W. Steinberg showed in 1957that G has an irreducible complex representation whose degreeequals the p-part of |G| [11]. This representation, now knownas the Steinberg representation, has remarkable properties,which reflect the structure of G, and there have been many researchpapers devoted to its study. The module constructed in [11]is in fact a right ideal in the integral group ring ZG of G,and is thus a ZG-lattice, which we propose to call the Steinberglattice of G. It should be noted that lattices not integrallyisomorphic to the Steinberg lattice may also afford the Steinbergrepresentation, and such lattices may differ considerably intheir properties compared with the Steinberg lattice.     

On the vanishing of subspaces of alternating bilinear forms

Given a field F and integer n≥3, we introduce an invariant s_n (F) which is defined by examining the vanishing of subspaces of alternating bilinear forms on 2-dimensional subspaces of vector spaces. This invariant arises when we calculate the largest dimension of a subspace of n?×?n skew-symmetric matrices over F which contains no elements of rank 2. We show how to calculate s_n (F) for various families of field F, including finite fields. We also prove the existence of large subgroups of the commutator subgroup of certain p-groups of class 2 which contain no non-identity commutators.     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

APN permutations on and Costas arrays

We study PN and APN functions over the integers modulo n. We give some construction techniques based on Costas arrays, which allow us to construct APN permutations on where p is a prime. Although PN permutations do not exist, one set of our functions is very close to being a set of PN permutations.     

Contractible subgraphs and Morita equivalence of graph -algebras

In this paper we describe an operation on directed graphs which produces a graph with fewer vertices, such that the -algebra of the new graph is Morita equivalent to that of the original graph. We unify and generalize several related constructions, notably delays and desingularizations of directed graphs.

     

Construction of p-1 Irreducible Modules with Fundamental Highest Weight for the Symplectic Group in Characteristic p

Let K be a field and let V be a vector space of dimension 2mover K. Let V denote the exterior algebra of V and ^kV its kthexterior power for 0k2m. Let f be a non-degenerate alternatingbilinear form defined on VxV. The symplectic group Sp_2m(K) isthe group of all isometries of f and it acts as a group of vectorspace automorphisms on ^kV. In the case that K is algebraicallyclosed and 1km, it is known that ^kV contains a composition factorcorresponding to the fundamental weight _k of a root system oftype C_m. We shall refer to the irreducible module for Sp_2m(K)given by this composition factor as a fundamental module.     

Cartan invariants and defect zero elements

In this note we prove relations between Cartan invariants in characteristc p and p defect zero elements. Exploiting duality arguments in characteristic two we show that the existence of real 2-defect zero elements is equivalent to the existence of odd diagonal entries in the Cartan matrix corresponding to real-valued Brauer characters. In addition, we present an independent and elementary proof of Theorem 3.3 part (iii) in [2]. Received: 9 January 2006