A Littlewood-Richardson Rule for Forest Polynomials via the Schubert Bialgebra

Abstract

The forest polynomials $\mathfrak{P}_a$ of Nadeau-Tewari form a $\mathbb{Z}$-basis of $\mathbb{Z}[x_1, x_2, \dots]$ whose role for the cohomology of the quasisymmetric flag variety parallels that of Schubert polynomials for the classical flag variety. Nonnegativity of the structure constants $β^c_{a,b}$ in $\mathfrak{P}_a \mathfrak{P}_b = \sum_c β^c_{a,b} \mathfrak{P}_c$ is known, but no Littlewood-Richardson-style enumerative rule has been available. We give such a rule: $β^c_{a,b}$ counts pairs of forest RC graphs of forest-codes $a$ and $b$ whose lift product lands on a forest RC graph of forest-code and weight both equal to $c$. The same rule descends to the cup product on $H^\bullet(QFl_n)$. The proof introduces a Schubert bialgebra $\mathcal{A}$ and lifts the multiplication on its graded dual $\mathcal{D}$ to a product on a free abelian group $\mathcal{B}RC$ of bounded RC graphs; the same machinery yields enumerative LR rules for the dual Schubert, dual key, dual forest, and dual slide bases of $\mathcal{D}$.