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October 8, 2025Open Access

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

Key Points

A new formula for littlewood-richardson coefficients simplifies calculations in combinatorial contexts, particularly for temperley-lieb immanants.
The formula draws connections to peelable tableaux and follows the framework of the remmel-whitney rule, enhancing computability.
Bender-knuth involutions are also linked, revealing deeper symmetry characteristics in littlewood-richardson coefficients that may assist future research.
Applications of this work include proving cases of the schur log-concavity conjecture, which broadens the implications of these findings.

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.

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