Pattern-avoiding polytopes and Cambrian lattices

In 2017, Davis and Sagan found that a pattern-avoiding Birkhoff subpolytope and an order polytope have the same normalized volume. They ask whether the two polytopes are unimodularly equivalent. We give an affirmative answer to a generalization of this question. For each Coxeter element c in the symmetric group, we define a pattern-avoiding Birkhoff subpolytope, and an order polytope of the heap poset of the c-sorting word of the longest permutation. We show the two polytopes are unimodularly equivalent. As a consequence, we show the normalized volume of the pattern-avoiding Birkhoff subpolytope is equal to the number of the longest chains in a corresponding Cambrian lattice. In particular, when c = s_1s_2…s_{n-1}, this resolves the question by Davis and Sagan. This talk is based on ongoing joint work with E. Banaian, S. Chepuri and E. Gunawan.