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arXiv · 2504.07505

$c$-Birkhoff polytopes

Esther Banaian·Sunita Chepuri·Emily Gunawan·Jianping Pan
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Abstract

In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan's question, in this paper we define a pattern-avoiding Birkhoff polytope called a $c$-Birkhoff polytope for each Coxeter element $c$ of the symmetric group. We then show that the $c$-Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the $c$-sorting word of the longest permutation. When $c=s_1s_2\dots s_{n}$, this result recovers an affirmative answer to Davis and Sagan's question. Another consequence of this result is that the normalized volume of the $c$-Birkhoff polytope is the number of the longest chains in the (type A) $c$-Cambrian lattice.

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Esther Banaian, Sunita Chepuri, Emily Gunawan, Jianping Pan. 2025-04-10. $c$-Birkhoff polytopes. https://doi.org/10.5802/alco.472

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