The invariant factors of the incidence matrices of points and subspaces in 𝑃𝐺(𝑛,𝑞) and 𝐴𝐺(𝑛,𝑞)

Abstract

Abstract. We determine the Smith normal forms of the incidence matrices of points and projective (r − 1)-dimensional subspaces of PG(n, q) and of the incidence matrices of points and r-dimensional affine subspaces of AG(n, q) for all n, r, and arbitrary prime power q.

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“…Yet that is exactly what we will do. The information that we need about the elementary divisors of A r,1 and A 1,s we obtain from [2]. A 1,s and a right SNF basis for A r,1 .…”

“…(1) The field k in [1] is actually an algebraic closure of F q , but (as observed in [2]) it follows from [1, Theorem A] that all kG-submodules of k L 1 are simply scalar extensions of F q G-modules, and therefore [1, Theorem A] is also true over our field F ∼ = F q n+1 . This observation also permits us to make use of certain results from [2], where the field is F q .…”

“…(2) It should also be noted that the incidence relation considered in [1,7,2] is nonzero intersection (i.e., the complementary relation where two subspaces are incident if and only if their intersection is nontrivial). If A r,s is the corresponding incidence matrix for nonzero intersection, then we have…”

“…Therefore the (p-adic) Smith normal forms of A r,s and A r,s can differ only with respect to where they map 1. This accounts for the extra term appearing in the calculation of p-ranks in [1,7,2].…”

The permutation action of finite symplectic groups of odd characteristic on their standard modules

“…The incidence matrices between P and flats of PG(2m − 1, q) have been studied extensively over the past forty years. See for example, [14,6,5,1,7] for F pranks of these matrices, and [12,2] for their Smith normal forms. The study of p-ranks of these incidence matrices led the authors of [1] to investigate the submodule lattices of the spaces k[P ] and k[V ] of k-valued functions on P and V respectively, viewed as permutation modules for the general linear group GL(V ).…”