Gelfand-Tsetlin polytopes and the integer decomposition property

TL;DR

This paper characterizes when Gelfand-Tsetlin polytopes are integral and explores their integer decomposition property, providing a partial ordering that supports a conjecture about their structure for various shapes.

Contribution

It completely characterizes integrality of Gelfand-Tsetlin polytopes for standard Young tableaux and introduces a partial order to analyze their integer decomposition property.

Findings

Gelfand-Tsetlin polytopes are integral for certain shapes

Integral Gelfand-Tsetlin polytopes are compressed

Partial order helps understand integrality and decomposition property

Abstract

Let $P$ be the Gelfand--Tsetlin polytope defined by the skew shape $\lambda/\mu$ and weight $w$. In the case corresponding to a standard Young tableau, we completely characterize for which shapes $\lambda/\mu$ the polytope $P$ is integral. Furthermore, we show that $P$ is a compressed polytope whenever it is integral and corresponds to a standard Young tableau. We conjecture that a similar property hold for arbitrary $w$, namely that $P$ has the integer decomposition property whenever it is integral. Finally, a natural partial ordering on GT-polytopes is introduced that provides information about integrality and the integer decomposition property, which implies the conjecture for certain shapes.