Journal of the Mathematical Society of Japan

Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group

Taro SUZUKI and Tatsuru TAKAKURA

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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.

Article information

J. Math. Soc. Japan, Volume 61, Number 3 (2009), 921-969.

First available in Project Euclid: 30 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 17B10: Representations, algebraic theory (weights) 53D20: Momentum maps; symplectic reduction

root system representation tensor product multiplicity coadjoint orbit symplectic quotient


SUZUKI, Taro; TAKAKURA, Tatsuru. Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group. J. Math. Soc. Japan 61 (2009), no. 3, 921--969. doi:10.2969/jmsj/06130921.

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