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 $\mathcal{F}_{\hat{G}_B(\nu)}$ 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 M\'esz\'aros' 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.