The equivariant Ehrhart theory of polytopes with order-two symmetries

Clarke, Oliver; Higashitani, Akihiro; Kölbl, Max

doi:10.1090/proc/16473

The equivariant Ehrhart theory of polytopes with order-two symmetries
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by Oliver Clarke, Akihiro Higashitani and Max Kölbl;

Proc. Amer. Math. Soc. 151 (2023), 4027-4041

DOI: https://doi.org/10.1090/proc/16473

Published electronically: June 6, 2023

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Abstract:

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant $H^\ast$-polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycle graphs and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope.

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