On the Subdivision Algebra for the Polytope \(\mathcal {U}_{I,\overline{J}}\)

The polytopes \(\mathcal {U}_{I,\overline{J}}\) were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of \((I,\overline{J})\)-Tamari lattices. They observed a connection between certain \(\mathcal {U}_{I,\overline{J}}\) and acyclic root polytopes, and wondered if Mészáros’ subdivision algebra can be used to subdivide all \(\mathcal {U}_{I,\overline{J}}\). We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that \(\mathcal {U}_{I,\overline{J}}\) is integrally equivalent to a flow polytope that can be subdivided using the subdivision algebra. Alternatively, we find a suitable projection of \(\mathcal {U}_{I,\overline{J}}\) to an acyclic root polytope which allows subdivisions of the root polytope to be lifted back to \(\mathcal {U}_{I,\overline{J}}\). As a consequence, this implies that subdivisions of \(\mathcal {U}_{I,\overline{J}}\) can be obtained with the algebraic interpretation of using reduced forms of monomials in the subdivision algebra. In addition, we show that the \((I,\overline{J})\)-Tamari complex can be obtained as a triangulated flow polytope.