The commuting complex of the symmetric group with bounded number of $p$-cycles

TL;DR

This paper studies a filtration of the commuting complex of elements of order p in symmetric groups, showing that each level becomes highly acyclic as n grows, using FI-modules in the proof.

Contribution

It introduces a new filtration of the commuting complex based on the number of disjoint p-cycles and proves high acyclicity using FI-modules.

Findings

Each filtration level becomes highly acyclic as n increases

The approach employs FI-modules to establish acyclicity

Provides a new perspective on the structure of commuting complexes

Abstract

For a fixed prime $p$, we consider a filtration of the commuting complex of elements of order $p$ in the symmetric group $\mathfrak{S}_n$. The filtration is obtained by imposing successively relaxed bounds on the number of disjoint $p$-cycles in the cycle decomposition of the elements. We show that each term in the filtration becomes highly acyclic as $n$ increases. We use $\mathbf{FI}$-modules in the proof.