2022/05/21 by von Bell, Matias, Yip, Martha
#05E45 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.10594
The polytopes U_I,J were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of (I,J)-Tamari lattices. They observed a connection between certain U_I,J and acyclic root polytopes, and wondered if Mészáros' subdivision algebra can be used to subdivide all U_I,J. We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that U_I,J is integrally equivalent to a flow polytope that can be subdivided using the subdivision algebra. Alternatively, we find a suitable projection of U_I,J to an acyclic root polytope which allows subdivisions of the root polytope to be lifted back to U_I,J. As a consequence, this implies that subdivisions of U_I,J can be obtained with the algebraic interpretation of using reduced forms of monomials in the subdivision algebra. In addition, we show that the (I,J)-Tamari complex can be obtained as a triangulated flow polytope.