A Littlewood-Richardson Rule for Forest Polynomials

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 $H^{\bullet }({\mathrm{QFl}}_n)$. The proof introduces a squash product $⊠$ on RC graphs, which on full Grassmannian RC graphs of a fixed height recovers the plactic monoid, a multiplactic algebra ${\mathcal{M}}_n$ that models Schubert polynomial multiplication, and a lift product $\ast$ on RC graphs whose quasi-crystal structure, in the framework of Cain-Guilherme-Malheiro, transports the Nadeau-Tewari forest equivalence through the computation. The row-decomposition machinery on RC graphs that underlies these constructions is developed in a companion paper.

arXiv comment28 pages. v2: the Schubert bialgebra material of v1 has been split off into the companion paper arXiv:2610.22582; this version contains the forest Littlewood-Richardson rule, reorganized from what was the final section of v1, with the needed background collected in Section 2. Title changed