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arXiv · 2206.02276

Restricted Birkhoff polytopes and Ehrhart period collapse

Per Alexandersson·Sam Hopkins·Gjergji Zaimi
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Abstract

We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the "longest increasing subsequence" have Ehrhart quasi-polynomials which are honest polynomials, even though they are just rational polytopes in general. We do this by defining a continuous, piecewise-linear bijection to a certain Gelfand-Tsetlin polytope. This bijection is not an integral equivalence but it respects lattice points in the appropriate way to imply that the two polytopes have the same Ehrhart (quasi-)polynomials. In fact, the bijection is essentially the Robinson-Schensted-Knuth correspondence.

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Per Alexandersson, Sam Hopkins, Gjergji Zaimi. 2022-06-05. Restricted Birkhoff polytopes and Ehrhart period collapse. https://doi.org/10.1007/s00454-023-00611-z

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