## Journal of the Mathematical Society of Japan

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

#### Abstract

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

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

Dates
First available in Project Euclid: 30 July 2009

https://projecteuclid.org/euclid.jmsj/1248961482

Digital Object Identifier
doi:10.2969/jmsj/06130921

Mathematical Reviews number (MathSciNet)
MR2552919

Zentralblatt MATH identifier
1179.22016

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1248961482

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