Andrés R. Vindas Meléndez, University of Kentucky

Fixed Subpolytopes of the Permutahedron 

Abstract: 

Motivated by the generalization of Ehrhart theory with group actions, this project makes progress towards obtaining the equivariant Ehrhart theory of the permutahedron. The fixed subpolytopes of the permutahedron are the polytopes that are fixed by acting on the permutahedron by a permutation. We prove some general results about the fixed subpolytopes. In particular, we compute their dimension, show that they are combinatorially equivalent to permutahedra, provide hyperplane and vertex descriptions, and prove that they are zonotopes. Lastly, we obtain a formula for the volume of these fixed subpolytopes, which is a generalization of Richard Stanley's result of the volume for the standard permutahedron. This is joint work with Federico Ardila (San Francisco State) and Anna Schindler (University of Washington).

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