Homology of matching complexes and representations of symmetric groups

TL;DR

The paper computes the homology of the matching complex for the complete hypergraph on $n$ vertices via shellings, obtaining $\tilde{H}_q(X(n),\mathbb{Z}) \cong \mathbb{Z}^{\beta(n,q)}$ with $\beta(n,q) = {n \brace q+1}_{2}$ (partitions of $[n]$ into $q+1$ blocks of size at least $2$) and identifying the $S_n$-action as a sum of induced modules from normalisers of standard Young subgroups. It then analyzes the resulting representations, showing that $\tilde{H}_q(X(n))$ decomposes as $\bigoplus_{\lambda} V(\lambda)\uparrow^{S_n}_{N_{\lambda}}$, with a Kostka-number specialization when $\lambda$ has distinct parts. Beyond this concrete case, the paper develops a general framework for fibre-closed families of simplicial complexes using FS$^{\text{op}}$-modules, proving an asymptotic representation-stability theorem: for fixed $q$, Betti numbers grow as a sum of polynomial-times-exponential terms, the $S_n$-modules stabilize with Specht components of bounded length, multiplicities are quasi-polynomial in $n$, and a finite, computable character-polynomial description governs the action, with torsion uniformly bounded. These results provide a broad, principled view of how homological and representation-theoretic structures behave in families of complexes under symmetric-group actions, with implications for understanding growth, stability, and torsion across combinatorial topologies.

Abstract

We compute the homology of the matching complex $M(Γ)$, where $Γ$ is the complete hypergraph on $n\geq 2$ vertices, and analyse the $S_n$-representations carried by this homology. These results are achieved using standard techniques in combinatorial topology, such as the theory of shellings. We then broaden the scope to the larger class of fibre-closed families of simplicial complexes and consider these through the lens of representation stability. This allows us to prove a number of results of an asymptotic nature, such as an analysis of the growth of Betti numbers and the kinds of irreducible $S_n$-representations that appear.

Theorems & Definitions (12)

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