Symmetry, Representations, and Invariants pp 363-385 | Cite as

# Branching Laws

## Abstract

Since each classical group *G* fits into a descending family of classical groups, the irreducible representations of *G* can be studied inductively. This gives rise to the *branching problem*: Given a pair *G* ⊃ *H* of reductive groups and an irreducible representation *π* of *G*, find the decomposition of *π*│_{ H } into irreducible representations. In this chapter we solve this problem for the pairs **GL**(*n,* \(\mathbb{C}\)) ⊃ **GL**(*n* -1,\(\mathbb{C}\)), **Spin**(*n,* \(\mathbb{C}\)) ⊃ **Spin**(*n* - 1,\(\mathbb{C}\)), and **Sp**(*n,* \(\mathbb{C}\)) ⊃ **Sp**(*n* -1,\(\mathbb{C}\)). We show that the representations occurring in *π*│_{ H } are characterized by a simple interlacing condition for their highest weights. For the first and second pairs the representation *π*│_{ H } is multiplicity-free; in the symplectic case the multiplicities are given in terms of the highest weights by a product formula. We prove all these results by a general formula due to Kostant that expresses branching multiplicities as an alternating sum over the Weyl group of *G* of a suitable partition function. In each case we show that this alternating sum can be expressed as a determinant. The explicit evaluation of the determinant then gives the branching law.

## Keywords

Partition Function High Weight Irreducible Representation Classical Group Positive Root## Preview

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