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c-Birkhoff polytopes
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DOI：10.5802/alco.472.png)
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Abstract
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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 = s1s2 ... sn, 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.
Keywords:
Birkhoff polytope
order polytope
heap
Cambrian lattice
c-singleton
Journal
A
IF:
0.9
Papers:
13
Citations:
0
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Cited Papers
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