The Equivariant Volumes of the Permutahedron

Abstract

We prove that if $$\sigma $$ is a permutation of $$S_n$$ with m cycles of lengths $$l_1, \ldots , l_m$$ , the subset of the permutahedron $$\Pi _n$$ fixed by the natural action of $$\sigma $$ is a polytope with volume $$n^{m-2} \gcd (l_1, \ldots , l_m)$$ .