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

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

2008-08-22

arxiv.org/abs/0808.3114v1

Mathematics

Group Theory

25 pages

Scientific paper

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 $p=3$.

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