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 $n\equiv 1\mathrm{mod}p$. Our conjecture follows from work of Bouc when $p=2$, and we prove the conjecture when $p=3$.

25 pages

References in corpus (1)

• Exact Sequences for the Homology of the Matching Complex