THE PERMUTATION REPRESENTATION OF Sp(2m,Fp) ACTING ON THE VECTORS OF ITS STANDARD MODULE

This paper studies the permutation representation of a finite symplectic group over a prime field of odd characteristic on the vectors of its standard module. The submodule lattice of this permutation module is determined. The results yield additive formulae for the p-ranks of various incidence matrices arising from the finite symplectic spaces. Introduction In this paper, we study the action of the symplectic group Sp(2m, p) on the set of vectors in its standard module. The composition factors of this permutation module have been known for some time ([9], [12], [13]) and so the problem we address here is that of describing the submodule lattice. This turns out to be quite similar to the known structure of this module under the action of the general linear group (See [3] and references cited there). This structural information yields additive formulae for the p-ranks of the incidence matrices between points and isotropic subspaces of fixed dimension in (2m−1)-dimensional projective space over Fp. This generalizes recent work [5] of de Caen and Moorhouse, who worked out the p-rank of the point-line incidence when m = 2. I wish to thank Eric Moorhouse and Alex Zalesskii for fruitful discussions and for supplying me with copies of their work, out of which this paper grew. §1. Functions on a finite vector space 1.1. Let p be an odd prime and let V be a 2m-dimensional Fp-vector space with a nonsingular alternating bilinear form 〈−,−〉. We shall assume m ≥ 2 to avoid trivial exceptions. We fix a symplectic basis e1, . . . em, fm, . . . f1 and corresponding coordinates X1, . . . , Xm, Ym, . . . Y1 so that 〈ei, fj〉 = δij . Supported by NSF grant DMS9701065 Typeset by AMS-TEX 1 Let k be an algebraic closure of Fp and let A = k[X1, . . . , Xm, Ym, . . . , Y1]/(Xi p −Xi, Yi p − Yi) m i=1 (1) be the ring of functions on V . This is the principal object of our study. 1.2. Structure of A as a kGL(V )-module. The the action of GL(V ) on A is induced from its action on the polynomial ring k[X1, . . . , Xm, Ym, . . . , Y1] through linear substitutions of the variables. The kGL(V )-module structure of A is well known; we will give a brief description. When we factor out by the inhomogeneous ideal (Xi p −Xi, Yi p − Yi) m i=1 the grading on k[X1, . . .Xm, Ym, . . . , Y1] is destroyed, leaving only a filtration {Fe} 2m(p−1) e=0 , where Fe = Image in A of polynomials of degree ≤ e, (2) and a Z/(p− 1)Z-grading (from the action of the scalar matrices) A = ⊕ d=0A[d], (3) where A[d] is the image of all homogeneous polynomials of degree congruent to d modulo p− 1. Denote by S(e) the component of degree e in the graded ring S = k[X1, . . . , Xm, Y1, . . . , Ym]/(Xi , Yi )i=1. (4) Here e ranges from 0 to 2m(p− 1). The dimension of S(e) is