The equivariant volumes 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 $l_1,\dots ,l_m$, then the subpolytope of ${\Pi }_n$ fixed by $\sigma$ has normalized volume $n^{m-2}\gcd (l_1,\dots ,l_m)$.

13 pages, 2 figures