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

TL;DR

This paper introduces a new combinatorial formula for Littlewood--Richardson coefficients using peelable tableaux, enabling efficient computation and proving a special case of Schur log-concavity.

Contribution

It presents a novel tableau-based rule for calculating Littlewood--Richardson coefficients and demonstrates its compatibility with symmetry and log-concavity conjectures.

Findings

New formula for Littlewood--Richardson coefficients

Efficient computation for Temperley--Lieb immanants

Proof of a special case of Schur log-concavity

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.