The Schubert bialgebra
Abstract
We introduce a Schubert bialgebra π=β¨nβ₯ 0β€[x1,β¦,xn], in which polynomials in different numbers of variables multiply to zero and the coproduct splits the variables into two consecutive blocks, and study its graded dual π, whose product is concatenation of weak compositions. The Schubert polynomials Sun form a basis of π; we give an explicit formula for the dual Schubert basis Ξun of π via a transition formula to an elementary symmetric basis, and show that the Schubert structure constants cu,vw are the coefficients of the coproduct of π in the dual Schubert basis. The product of π is then lifted to an associative product sqcup on the free abelian group β¬βπ of bounded RC graphs, built from a transition map π΅ on RC graphs that realizes the Lascoux-SchΓΌtzenberger transition formula and generalizes Little's bumping algorithm. Quotienting β¬βπ by ideals coming from Demazure crystal equivalence, Nadeau-Tewari forest equivalence, and quasi-Yamanouchi equivalence yields manifestly positive Littlewood-Richardson rules for the dual Schubert, dual key, dual forest, and dual slide bases of π. Read on π, these are branching rules: enumerative formulas for splitting a Schubert, key, forest, or slide polynomial across two blocks of variables, and in the Schubert case they compute the Levi branching coefficients of Ressayre-Richmond for GLpΓGLqβͺGLp+q.
Related
- Differential perm bialgebras, differential perm Yang-Baxter equation, and their applications in related bialgebra structures
- Universal PlΓΌcker coordinates for the Wronski map and positivity in real Schubert calculus
- Signed puzzles for Schubert coefficients
- Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions
- Normality of closures of conormal bundles to Schubert cells for GLn
BibTeX
@misc{samuel2026schubert,
title = {{The Schubert bialgebra}},
author = {Matthew J. Samuel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22582},
eprint = {2610.22582},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22582},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Schubert bialgebra
AU - Matthew J. Samuel
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22582
UR - https://arxiv.org/abs/2610.22582
AB - We introduce a Schubert bialgebra π=β¨_nβ₯ 0β€[x_1,β¦,x_n], in which polynomials in different numbers of variables multiply to zero and the coproduct splits the variables into two consecutive blocks, and study its graded dual π, whose product is concatenation of weak compositions. The Schubert polynomials S_u^n form a basis of π; we give an explicit formula for the dual Schubert basis Ξ_u^n of π via a transition formula to an elementary symmetric basis, and show that the Schubert structure constants c_u,v^w are the coefficients of the coproduct of π in the dual Schubert basis. The product of π is then lifted to an associative product sqcup on the free abelian group β¬βπ of bounded RC graphs, built from a transition map π΅ on RC graphs that realizes the Lascoux-SchΓΌtzenberger transition formula and generalizes Little's bumping algorithm. Quotienting β¬βπ by ideals coming from Demazure crystal equivalence, Nadeau-Tewari forest equivalence, and quasi-Yamanouchi equivalence yields manifestly positive Littlewood-Richardson rules for the dual Schubert, dual key, dual forest, and dual slide bases of π. Read on π, these are branching rules: enumerative formulas for splitting a Schubert, key, forest, or slide polynomial across two blocks of variables, and in the Schubert case they compute the Levi branching coefficients of Ressayre-Richmond for GL_pΓGL_qβͺGL_p+q.
ER -