Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity

2025/05/31 by Chau Nguyen, Nguyen, Chau, Nguyễn Thái Sơn +3
Mathematics · #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2506.00349

Abstract

We give a new formula for the Littlewood--Richardson coefficients in terms of peelable tableaux compatible with shuffle tableaux, in the same fashion as Remmel--Whitney rule. This gives an efficient way to compute generalized Littlewood--Richardson coefficients for Temperley--Lieb immanants of Jacobi--Trudi matrices. We will also show that our rule behaves well with Bender--Knuth involutions, recovering the symmetry of Littlewood--Richardson coefficients. As an application, we use our rule to prove a special case of a Schur log-concavity conjecture by Lam--Postnikov--Pylyavskyy.

Citations

• Temperley-Lieb Crystals
• The Cactus Group Property for Ordinal Sums of Disjoint Unions of Chains
• Bender-Knuth involutions on linear extensions of posets
• The Berenstein-Kirillov group and cactus groups
• Alcoved Polytopes II
• On products of sl_n characters and support containment
• Schur positivity and Schur log-concavity
• Alcoved Polytopes I
• Some positive differences of products of Schur functions
• Eigenvalues, singular values, and Littlewood-Richardson coefficients
• Tensor product multiplicities, canonical bases and totally positive varieties
• The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture
• Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties

Related

• Equivariant Littlewood-Richardson Skew Tableaux
• Temperley-Lieb Crystals
• Littlewood--Richardson coefficients and integrable tilings
• Shifted tableaux and products of Schur's symmetric functions
• Constructed tableaux and a new shifted Littlewood-Richardson rule