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 ${\beta }_{a,b}^c$ in ${\mathfrak{P}}_a{\mathfrak{P}}_b={\sum }_c{\beta }_{a,b}^c{\mathfrak{P}}_c$ is known, but no Littlewood-Richardson-style enumerative rule has been available. We give such a rule: ${\beta }_{a,b}^c$ 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 $H^{\bullet }(QF{l}_n)$. 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 $\mathcal{D}$.

58 pages, 6 figures