Incidence Homology of Finite Projective Spaces

Details Incidence Homology of Finite Projective Spaces Incidence Homology of Finite Projective Spaces

2011-10-23

arxiv.org/abs/1110.5031v3

Mathematics

Combinatorics

The paper has 16 pages with 2 figures

Scientific paper

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.

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