A subdivision algebra for a product of two simplices via flow polytopes

Abstract

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 ${\mathcal{F}}_{{\hat{G}}_B(\nu )}$ arising from an acyclic graph where bidirectional edges are permitted. We show that the flow polytope ${\mathcal{F}}_{{\hat{G}}_B(\nu )}$ admits a subdivision dual to a $w$-simplex, where $w$ is the number of valleys in the path $\bar{\nu }=E\nu N$. 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 ${\mathcal{F}}_{{\hat{G}}_B(\nu )}$ 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.

19 pages, 10 figures