Title: Polytopes from heaps
Abstract: The Birkhoff polytope is the convex hull of all permutation matrices in a fixed symmetric group. We study certain subpolytopes of the Birkhoff polytope whose vertex sets coincide with prefixes of a set of commutation-related reduced words. The vertices are indexed by order ideals (downwards-closed sets) of a certain poset called a heap. The set of order ideals of a poset also labels the vertices of its order polytope. Based on a result and question by Davis and Sagan, we ask: for a fixed heap, when are these two polytopes, namely the ``heap Birkhoff subpolytope'' and the order polytope, integrally equivalent? We discuss a class of heaps for which the phenomenon is true and progress towards understanding when the polytopes fail to be equivalent. Time-permitting, we also discuss how to consider the question in type B. This is based on current joint work with Sunita Chepuri, Emily Gunawan, and Jianping Pan, part of which is available at arXiv:2504.07505.
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