On the Subdivision Algebra for the Polytope U_I,J¯

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.