The equivariant volumes of the permutahedron

TL;DR

This paper investigates the symmetric group action on the permutahedron and derives a formula for the volume of fixed subpolytopes based on permutation cycle structure.

Contribution

It provides a novel explicit formula for the normalized volume of fixed subpolytopes of the permutahedron under symmetric group actions.

Findings

Volume of fixed subpolytope depends on permutation cycle structure.

Normalized volume equals n^{m-2} times the gcd of cycle lengths.

Results connect permutation cycles with geometric properties of the permutahedron.

Abstract

We consider the action of the symmetric group $S_n$ on the permutahedron $\Pi_n$. We prove that if $\sigma$ is a permutation of $S_n$ which has $m$ cycles of lengths $l_1, \ldots, l_m$, then the subpolytope of $\Pi_n$ fixed by $\sigma$ has normalized volume $n^{m-2} \gcd(l_1, \ldots, l_m)$.