Positivity for a t-analogue of refined Littlewood-Richardson rules
Abstract
In 2010, Haglund, Luoto, Mason and van Willigenburg proved that the product of a Schur function sλ and a Demazure atom Aγ expands positively in terms of Demazure atoms, and interpreted the coefficients as the number of Littlewood-Richardson skew skyline tableaux of a given shape and a filling type. Extending this approach, they further established the positive expansion for the product of a Schur function sλ and a quasisymmetric Schur function Sα, or of a Schur function sλ and a Demazure character κγ. In particular, the latter provides a refinement of the classical Littlewood-Richardson rule. We present a natural t-analogue of these positive expansions by deriving the expansion for a product of a Hall-Littlewood polynomial Pλ(t) and a t-Demazure atom Eγ(t) in terms of t-Demazure atoms. The corresponding coefficients are ℕ [t,1−t]-positive; that is, they are polynomials with nonnegative integer coefficients in t and 1−t, and they are interpreted in terms of standard Young tableaux. We take the approach of connecting the multiplication coefficients with the LLT polynomials, which differs from the combinatorial insertion arguments of Haglund, Luoto, Mason and van Willigenburg.
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BibTeX
@misc{du2026positivity,
title = {{Positivity for a $t$-analogue of refined Littlewood-Richardson rules}},
author = {Jiaming Du and Emma Yu Jin and Xiaowei Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35708},
eprint = {2610.35708},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35708},
}
RIS
TY - UNPB
TI - Positivity for a t-analogue of refined Littlewood-Richardson rules
AU - Jiaming Du
AU - Emma Yu Jin
AU - Xiaowei Lin
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.35708
UR - https://arxiv.org/abs/2610.35708
AB - In 2010, Haglund, Luoto, Mason and van Willigenburg proved that the product of a Schur function s_λ and a Demazure atom A_γ expands positively in terms of Demazure atoms, and interpreted the coefficients as the number of Littlewood-Richardson skew skyline tableaux of a given shape and a filling type. Extending this approach, they further established the positive expansion for the product of a Schur function s_λ and a quasisymmetric Schur function S_α, or of a Schur function s_λ and a Demazure character κ_γ. In particular, the latter provides a refinement of the classical Littlewood-Richardson rule. We present a natural t-analogue of these positive expansions by deriving the expansion for a product of a Hall-Littlewood polynomial P_λ(t) and a t-Demazure atom E_γ(t) in terms of t-Demazure atoms. The corresponding coefficients are ℕ [t,1−t]-positive; that is, they are polynomials with nonnegative integer coefficients in t and 1−t, and they are interpreted in terms of standard Young tableaux. We take the approach of connecting the multiplication coefficients with the LLT polynomials, which differs from the combinatorial insertion arguments of Haglund, Luoto, Mason and van Willigenburg.
ER -