Ehrhart positivity of Tesler polytopes and Berline-Vergne's valuation

DOI10.1007/S00454-022-00453-1arXiv1911.06291MaRDI QIDQ6329193

Fu Liu, Yonggyu Lee

Publication date: 14 November 2019

Abstract: For $$, the Tesler polytope $$ is the set of upper triangular matrices with non-negative entries whose hook sum vector is $$. Motivated by a conjecture of Morales', we study the questions of whether the coefficients of the Ehrhart polynomial of $e{s}_n(1,1,dots,1)$ are positive. We attack this problem by studying a certain function constructed by Berline-Vergne and its values on faces of a unimodularly equivalent copy of $e{s}_n(1,1,dots,1).$ We develop a method of obtaining the dot products appeared in formulas for computing Berline-Vergne's function directly from facet normal vectors. Using this method together with known formulas, we are able to show Berline-Vergne's function has positive values on codimension $2$ and $3$ faces of the polytopes we consider. As a consequence, we prove that the $3$rd and $4$th coefficients of the Ehrhart polynomial of $e{s}_n(1,dots,1)$ are positive. Using the Reduction Theorem by Castillo and the second author, we generalize the above result to all deformations of $e{s}_n(1,dots,1)$ including all the integral Tesler polytopes.

Mathematics Subject Classification ID

Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) (52B20) Dissections and valuations (Hilbert's third problem, etc.) (52B45)

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