Top homology of hypergraph matching complexes, p-cycle complexes and Quillen complexes of symmetric groups

Abstract

We investigate the representation of a symmetric group S n on the homology of its Quillen complex at a prime p. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of p-uniform hypergraph matching complexes. We conjecture an explicit formula for the representation of S n on the top homology group of the corresponding hypergraph matching complex when n ≡ 1 mod p . Our conjecture follows from work of Bouc when p = 2 , and we prove the conjecture when p = 3 .