The permutation action of finite symplectic groups of odd characteristic on their standard modules. (Q2470376)

Let \(k\) be a finite field of order \(q=p^t\) for some odd prime \(p\) and positive integer \(t\), and let \(V\) be a \(2m\)-dimensional vector space over \(k\) with the additional structure of a nonsingular alternating bilinear form. Let \(\text{PG}(2m-1,q)\) denote the projective geometry of \(V\), and let \(\text{W}(2m-1,q)\) denote the symplectic polar space whose flats are the flats of \(\text{PG}(2m-1,q)\) which are totally isotropic with respect to the form. Let \(P\) denote the set of points of \(\text{PG}(2m-1,q)\) (and hence also of \(\text{W}(2m-1,q)\)). The incidence matrices between \(P\) and flats of \(\text{PG}(2m-1,q)\) have been extensively studied over many years; in particular, in [\textit{M. Bardoe} and \textit{P. Sin}, J. Lond. Math. Soc., II. Ser. 61, No. 1, 58-80 (2000; Zbl 0961.20010)], the \(p\)-ranks of these matrices are obtained by studying the spaces \(k[V]\) and \(k[P]\) of \(k\)-valued functions on \(V\) and \(P\) as modules for the general linear group \(\text{GL}(V)\). In the paper under review, the analogous programme is carried out for the symplectic group \(\text{Sp(V)}\) of the form, yielding information about the incidence matrices between \(P\) and flats of \(\text{W}(2m-1,q)\). An important ingredient in this paper is the introduction of a special basis of \(k[V]\) consisting of so-called `symplectic basis functions'. The main results of the paper describe the structure of submodules generated by symplectic basis functions; these results, in turn, allow the authors to prove a symplectic analogue of Hamada's additive formula for \(p\)-ranks of incidence matrices [see \textit{N. Hamada}, J. Sci. Hiroshima Univ., Ser. A-I 32, 381-396 (1969; Zbl 0172.43203)]. A more detailed analysis in the case \(m=2\) is also presented, which gives a closed formula for the \(p\)-rank of the incidence matrix between points and lines of a symplectic generalized quadrangle over a field of odd order. Together with earlier results for the case of even characteristic, this completes the determination of the \(p\)-ranks for symplectic generalized quadrangles.