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Branching Laws

  • Roe Goodman
  • Nolan R. Wallach
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 255)

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 GH 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

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