THE ELEMENTARY DIVISORS OF THE INCIDENCE MATRICES OF POINTS AND LINEAR SUBSPACES IN P(Fp)

The elementary divisors of the incidence matrices between points and linear subspaces of fixed dimension in Pn(Fp) are computed. Introduction Let V be an (n+ 1)-dimensional vector space over Fp. Let Lr denote the set of r-dimensional subspaces of V . Then L1 is the set of points of the projective space P(V ) and Ln the set of hyperplanes. The group G = GL(V ) acts transitively on each of the sets Lr. Between any two of these sets 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 paper we shall be concerned with the cases in which one of the sets is L1. The incidence relation can be interpreted as the map Zr −→ Z1 (1) between the associated ZG-permutation modules which sends an r-subspace to the (formal) sum of the 1-subspaces it contains. This homomorphism has a finite cokernel and finding the elementary divisors of the incidence matrix is equivalent to finding a cyclic decomposition of the cokernel. The problem falls naturally into the two separate parts of describing the p-torsion and the p-torsion. The p-torsion can be obtained as a corollary of James’ theory [6] of crosscharacteristic modular representations of GL(n, q), where q is a power of p. It is a cyclic group of order p −1 p−1 , the number of 1-subspaces in an r-subspace. To see this consider the map ǫ : Z1 −→ Z sending each 1-subspace to 1. The image of the incidence map (1) is mapped by ǫ onto p −1 p−1 Z. The result will therefore follow if we show that the intersection of the image of (1) with Ker ǫ has index a Supported by NSF grant DMS9701065 Typeset by AMS-TEX 1 power of p in Ker ǫ. Since Ker ǫ is a pure subgroup of Z1 , this is equivalent to the statement that for each prime l 6= p, the reduction mod l of Ker ǫ is in the image of the reduction mod l of the map (1). It is this last fact which has been proved in [6, Theorem 13.3 and 11.1, Submodule Theorem]. The same argument works with p replaced by q, showing that the cokernel of the map from r-spaces to 1-spaces of a finite vector space over Fq is the product of a cyclic group of order (q − 1)/(q− 1) with a p-group. In this paper we concentrate on the p-torsion. Let di be the coefficient of t i(p−1) in the expansion of ( ∑p−1 j=0 t ). Explicitly,