A Subdivision Algebra for a Product of Two Simplices via Flow Polytopes

For a lattice path \(\nu \) from the origin to a point (a, b) using steps \(E=(1,0)\) and \(N=(0,1)\), we construct an associated flow polytope \({\mathcal {F}}_{{\widehat{G}}_B(\nu )}\) arising from an acyclic graph where bidirectional edges are permitted. We show that the flow polytope \({\mathcal {F}}_{{\widehat{G}}_B(\nu )}\) admits a subdivision dual to a \((w-1)\)-simplex, where w is the number of valleys in the path \({\overline{\nu }} = E\nu N\). Refinements of this subdivision can be obtained by reductions of a polynomial \(P_\nu \) in a generalization of Mészáros’ subdivision algebra for acyclic root polytopes where negative roots are allowed. Via an integral equivalence between \({\mathcal {F}}_{{\widehat{G}}_B(\nu )}\) and the product of simplices \(\Delta _a\times \Delta _b\), we thereby obtain a subdivision algebra for a product of two simplices. As a special case, we give a reduction order for reducing \(P_\nu \) that yields the cyclic \(\nu \)-Tamari complex of Ceballos, Padrol, and Sarmiento.