Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals

TL;DR

This paper uses quiver theory to analyze generalized Littlewood-Richardson coefficients for GL(n) branching rules, proving their saturation, non-vanishing conditions, and providing efficient computation methods.

Contribution

It introduces a quiver-theoretic framework for these coefficients, establishes their saturation property, and offers a polynomial-time algorithm for positivity testing.

Findings

Proved saturation of generalized Littlewood-Richardson coefficients.

Provided a polytopal description enabling efficient positivity computation.

Extended results to certain other generalized Littlewood-Richardson coefficients.

Abstract

Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in \cite{Chi08}, we use quiver theory to represent the generalized Littlewood-Richardson coefficients for the branching rule for the diagonal embedding of $\gl(n)$ as the dimension of a weight space of semi-invariants. Using this, we prove their saturation and investigate when they are nonzero. We also show that for certain partitions the associated stretched polynomials satisfy the same conjectures as single Littlewood-Richardson coefficients. We then provide a polytopal description of this multiplicity and show that its positivity may be computed in strongly polynomial time. Finally, we remark that similar results hold for certain other generalized Littlewood-Richardson coefficients.