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

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.

7 pages