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July, 2009 Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group
J. Math. Soc. Japan 61(3): 921-969 (July, 2009). DOI: 10.2969/jmsj/06130921


We consider asymptotic behavior of the dimension of the invariant subspace in a tensor product of several irreducible representations of a compact Lie group G . It is equivalent to studying the symplectic volume of the symplectic quotient for a direct product of several coadjoint orbits of G . We obtain two formulas for the asymptotic dimension. The first formula takes the form of a finite sum over tuples of elements in the Weyl group of G . Each term is given as a multiple integral of a certain polynomial function. The second formula is expressed as an infinite series over dominant weights of G . This could be regarded as an analogue of Witten's volume formula in 2-dimensional gauge theory. Each term includes data such as special values of the characters of the irreducible representations of G associated to the dominant weights.


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Taro SUZUKI. Tatsuru TAKAKURA. "Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group." J. Math. Soc. Japan 61 (3) 921 - 969, July, 2009.


Published: July, 2009
First available in Project Euclid: 30 July 2009

zbMATH: 1179.22016
MathSciNet: MR2552919
Digital Object Identifier: 10.2969/jmsj/06130921

Primary: 22E46
Secondary: 17B10 , 53D20

Keywords: coadjoint orbit , multiplicity , representation , root system , symplectic quotient , tensor product

Rights: Copyright © 2009 Mathematical Society of Japan

Vol.61 • No. 3 • July, 2009
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