A Littlewood-Richardson Rule for Forest Polynomials
Abstract
The forest polynomials Pa of Nadeau-Tewari form a ℤ-basis of ℤ[x1,x2,…] 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 βa,bc in Pa Pb = ∑c βa,bc Pc is known, but no Littlewood-Richardson-style enumerative rule has been available. We give such a rule: βa,bc 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•(QFln). The proof introduces a squash product boxtimes on RC graphs, which on full Grassmannian RC graphs of a fixed height recovers the plactic monoid, a multiplactic algebra ℳn that models Schubert polynomial multiplication, and a lift product * 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.
Related
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BibTeX
@misc{samuel2026littlewoodrichardson,
title = {{A Littlewood-Richardson Rule for Forest Polynomials}},
author = {Matthew J. Samuel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.23876},
eprint = {2606.23876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23876},
}
RIS
TY - UNPB
TI - A Littlewood-Richardson Rule for Forest Polynomials
AU - Matthew J. Samuel
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2606.23876
UR - https://arxiv.org/abs/2606.23876
AB - The forest polynomials P_a of Nadeau-Tewari form a ℤ-basis of ℤ[x_1,x_2,…] 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_a,b^c in P_a P_b = ∑_c β_a,b^c P_c is known, but no Littlewood-Richardson-style enumerative rule has been available. We give such a rule: β_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^•(QFl_n). The proof introduces a squash product boxtimes on RC graphs, which on full Grassmannian RC graphs of a fixed height recovers the plactic monoid, a multiplactic algebra ℳ_n that models Schubert polynomial multiplication, and a lift product * 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.
ER -