Unitary actions and equivariant volumes of symmetric edge polytopes

Abstract

The symmetric edge polytope ($\mathrm{SEP}$) of a finite simple graph $G$ is a centrally symmetric lattice polytope whose vertices are defined by the edges of the graph. Among the information encoded by these polytopes are the symmetries of the graph, which appear as symmetries of the polytope. We describe the rigid symmetries of these polytopes, and show that $\mathrm{SEP}$s are unitarily equivalent exactly when their associated graphs are isomorphic. We then find an explicit relationship between the relative volumes of the subsets of the symmetric edge polytope $\mathrm{SEP}$ fixed by the natural action of symmetric group elements and the symmetric edge polytopes of smaller graphs to which the subsets are linearly equivalent. We also provide a vertex description of the fixed polytopes and find a description of the symmetric edge polytopes to which they are equivalent, in terms of contractions of the graph $G$ induced by the cycle decompositions of the permutations under which the subsets are fixed. Specializations of our results provide equivalence and volume relationships for fixed polytopes of symmetric edge polytopes of complete graphs (equivalently, for fixed polytopes of root polytopes of type $A_n$), and describe the symmetry group of this family of polytopes.