Deformation cone of Tesler polytopes

Abstract

For $\mathbf{a}\in {\mathbb{R}}_{\ge 0}^n$, 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 ${\mathbf{a}}_0\in {\mathbb{R}}_{>0}^n$, 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 $\tes_{n}(\boldsymbol{a}_{0})$.