Incidence Homology of Finite Projective Spaces

TL;DR

This paper introduces a new family of combinatorial homology modules associated with finite projective spaces, providing a complete characterization of their structure and representations under GL(n,q).

Contribution

It constructs and fully characterizes a novel family of homology modules for finite projective spaces, including duality and representation-theoretic results.

Findings

Established a duality theorem for the homology modules.

Determined the modules' structure in terms of GL(n,q) irreducibles.

Provided a branching rule to analyze the modules' representations.

Abstract

Let F* be the finite field of q elements and let P(n,q) be the projective space of dimension n-1 over F*. We construct a family H^{n}_{k,i} of combinatorial homology modules associated to P(n,q) over a coefficient field F field of characteristic p_{0}>0 co-prime to q. As FGL(n,q)-representations the modules are obtained from the permutation action of GL(n,q) on the subspaces of F*^n. We prove a branching rule for H^{n}_{k,i} and use this rule to determine these homology representations completely. The main results are a duality theorem and the complete characterisation of H^{n}_{k,i} in terms of the standard irreducibles of GL(n,q) over F and applications.