Branching Laws

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


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.


Partition Function High Weight Irreducible Representation Classical Group Positive Root 
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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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