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{1248961482,
author = {SUZUKI, Taro and TAKAKURA, Tatsuru},
title = {Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group},
journal = {J. Math. Soc. Japan},
volume = {61},
number = {3},
year = {2009},
pages = { 921-969},
language = {en},
url = {http://dml.mathdoc.fr/item/1248961482}
}
SUZUKI, Taro; TAKAKURA, Tatsuru. Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group. J. Math. Soc. Japan, Tome 61 (2009) no. 3, pp. 921-969. http://gdmltest.u-ga.fr/item/1248961482/