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

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