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

TL;DR

This paper studies the symmetric group representations on the homology of Quillen complexes and hypergraph matching complexes, providing explicit formulas for small codimension and conjecturing the top homology case, with proofs for p=3.

Contribution

It derives explicit formulas for symmetric group actions on homology of hypergraph matching complexes and conjectures the top homology representation for certain cases, extending prior work.

Findings

Explicit formulas for small codimension homology representations.

Conjecture on the top homology representation when n ≡ 1 mod p.

Proof of the conjecture for p=3.

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