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 $\tes_{ n}(\boldsymbol{ a}_{ 0})$.
Categories: math.CO