Reduction for branching multiplicities

TL;DR

This paper extends known reduction formulas for branching coefficients in representation theory from the case where the Littlewood-Richardson coefficient is 1 to the case where it is 2, providing new insights into the structure of these coefficients.

Contribution

It introduces a new reduction formula for branching coefficients when the Littlewood-Richardson coefficient equals 2, expanding the understanding of tensor product restrictions.

Findings

Established a reduction formula for coefficient equal to 2

Analyzed properties of the branch divisor in degree 2 morphisms

Extended previous results from coefficient 1 to 2

Abstract

A reduction formula for the branching coefficients of tensor products of representations and more generally restrictions of representations of a semisimple group to a semisimple subgroup is proved in work by Knutson-Tao and Derksen-Weyman. This formula holds when the highest weights of the representations belong to a codimension 1 face of the Horn cone, which by work by Ressayre corresponds to a Littlewood-Richardson coefficient equal to 1. We prove a similar reduction formula when this coefficient is equal to 2, and show some properties of the class of the branch divisor corresponding to a generically finite morphism of degree 2 naturally defined in this context.