Relative Weyl Character formula, Relative Pieri formulas and Branching rules for Classical groups

TL;DR

This paper provides new proofs of classical branching rules for representations of certain classical groups, using a relative Weyl character formula and an inductive approach based on Pieri formulas.

Contribution

It introduces an alternative, division-based proof method for branching rules of classical groups, extending to cases involving Spin groups and reducing complex cases to simpler ones.

Findings

New proofs of classical branching rules for classical groups.

Extension of methods to Spin groups and reduction techniques.

Simplification of proofs using relative Weyl character and Pieri formulas.

Abstract

We give alternate proofs of the classical branching rules for highest weight representations of a complex reductive group $G$ restricted to a closed regular reductive subgroup $H$, where $(G,H)$ consist of the pairs $(GL(n+1),GL(n))$, $ (Spin(2n+1), Spin(2n)) $ and $(Sp(2n),Sp(2)\times Sp(2n-2))$. Our proof is essentially a long division. The starting point is a relative Weyl character formula and our method is an inductive application of a relative Pieri formula. We also give a proof of the branching rule for the case of $ (Spin(2n), Spin(2n-1))$, by a reduction to the case of $(GL(n),GL(n-1))$.