Deformation cone of Tesler polytopes

TL;DR

This paper investigates the deformation structure of Tesler polytopes, providing a new proof of their deformation relationships, calculating their deformation cone, and extending the analysis to related flow polytopes.

Contribution

It offers a new proof of Tesler polytopes' deformation relationships, computes their deformation cone, and characterizes which flow polytopes are deformations of a fixed Tesler polytope.

Findings

All Tesler polytopes with positive hook sums are deformations of a fixed Tesler polytope.

The deformation cone of a Tesler polytope is explicitly calculated.

Characterization of flow polytopes that are deformations of a given Tesler polytope.

Abstract

For $\boldsymbol{a} \in \R_{\geq 0}^{n}$, the Tesler polytope $\tes_{n}(\boldsymbol{a})$ is the set of upper triangular matrices with non-negative entries whose hook sum vector is $\ba$. We first give a different proof of the known fact that for every fixed $\boldsymbol{a}_{0} \in \mathbb{R}_{>0}^{n}$, all the Tesler polytopes $\tes_{n}(\boldsymbol{a})$ are deformations of $\tes_{n}(\boldsymbol{a}_{0})$. We then calculate the deformation cone of $\tes_{n}(\boldsymbol{a}_{0})$. In the process, we also show that any deformation of $\tes_{n}(\boldsymbol{a}_{0})$ is a translation of a Tesler polytope. Lastly, we consider a larger family of polytopes called flow polytopes which contains the family of Tesler polytopes and give a characterization on which flow polytopes are deformations of $\tes_{n}(\boldsymbol{a}_{0})$.