Polytopes and large counterexamples

TL;DR

This paper constructs large counterexamples to questions about order polytopes, especially Gelfand--Tsetlin polytopes, revealing limitations in current understanding and computational methods, and shows that hook multisets do not determine Ehrhart polynomials.

Contribution

It provides large, previously unknown counterexamples to natural questions about order polytopes and demonstrates that hook multisets are insufficient to determine Ehrhart polynomials.

Findings

Counterexamples are too large for brute-force discovery

Hook multisets do not determine Ehrhart polynomials

Challenges assumptions about order polytope invariants

Abstract

In this short note, we give large counterexamples to natural questions about certain order polytopes, in particular, Gelfand--Tsetlin polytopes. Several of the counterexamples are too large to be discovered via a brute-force computer search. We also show that the multiset of hooks in a Young diagram is not enough information to determine the Ehrhart polynomial for an associated order polytope. This is somewhat counter-intuitive to the fact that the multiset of hooks always determine the leading coefficient of the Ehrhart polynomial.