Branching polytopes for classical Lie algebras over $A_{n-1}$

TL;DR

This paper presents an inductive method to describe how classical Lie algebras branch over $A_{n-1}$, enabling labeling of highest weight vectors via lattice points of specific polytopes.

Contribution

It introduces a new inductive approach to describe branching rules for classical Lie algebras over $A_{n-1}$ using polytopes, building on Gornitskii's work.

Findings

Provides a labeling scheme for highest weight vectors using lattice points.

Uses string and Lusztig polytopes to describe module decompositions.

Offers an inductive framework for classical Lie algebra branching.

Abstract

We describe the branching of Lie algebras of classical type over $A_{n-1}$ using an inductive approach, which was motivated by the work of Gornitskii. This allows us to label the highest weight vectors of the modules occurring in the decomposition of the restriction of a finite-dimensional simple module to $A_{n-1}$ by lattice points of a string or a Lusztig polytope.