Polytopes and Large Counterexamples

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.

Keywords:

• polytopes
• counter-examples
• hook formula
• Ehrhart polynomial
• Kostka coefficient

2010 AMS Subject Classification:

• 52B12
• 05E10

Acknowledgments

The author would like to thank Greta Panova for pointing out reference [Pak 01] as well as for several insightful comments. We also thank Winfied Bruns for the computations in Normaliz.

Funding

This work was supported by the Knut and Alice Wallenberg Foundation (2013.03.07).