Flagged Littlewood-Richardson tableaux and branching rule for classical groups

TL;DR

This work introduces a novel, subtraction-free combinatorial formula for branching from $GL_n$ to $O_n$ using flagged Littlewood–Richardson tableaux and a spinor-crystal model for type $D_\infty$. By developing a separation algorithm that encodes Howe duality via a body/tail decomposition and sliding operations, the authors reduce branching multiplicities to indexed counts of LR tableaux with explicit flag bounds, recapturing Littlewood’s classical restriction in stable ranges. A key application is a new LR-based description of Lusztig $t$-weight multiplicities $K_{\mu 0}(t)$ for $B_n$ and $D_n$, together with a combinatorial model for generalized exponents of $\mathfrak{so}_n$ via distinguished tableaux. The results unify and extend prior non-orthogonal formulas, connect to Howe duality, and yield practical, subtraction-free combinatorial formulas for representation-theoretic branching problems in classical groups.

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 $K_{μ0}(t)$ of type $B_n$ and $D_n$ with highest weight $μ$ and weight $0$.