Flagged Littlewood-Richardson tableaux and branching rule for classical groups

TL;DR

This paper introduces a new combinatorial formula for the branching rule from general linear groups to orthogonal groups, utilizing flagged Littlewood-Richardson tableaux, and applies it to Lusztig t-weight multiplicities in types B and D.

Contribution

It generalizes Littlewood's restriction formula by incorporating flag conditions in tableaux, providing a novel combinatorial approach to branching rules and weight multiplicities.

Findings

New formula for GL_n to O_n branching rule

Combinatorial description using flagged Littlewood-Richardson tableaux

Explicit formula for Lusztig t-weight multiplicities in types B and D

Abstract

We give a new formula for the branching rule from ${\rm GL}_n$ to ${\rm O}_n$ generalizing the Littlewood's restriction formula. The formula is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish in a stable range. As an application, we give a combinatorial formula for the Lusztig $t$-weight multiplicity $K_{\mu 0}(t)$ of type $B_n$ and $D_n$ with highest weight $\mu$ and weight $0$.