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

• SUZUKI Taro

  Graduate School of Science and Engineering, Chuo University

• TAKAKURA Tatsuru

  Department of Mathematics, Chuo University

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.

Journal

• Journal of the Mathematical Society of Japan

  Journal of the Mathematical Society of Japan 61 (3), 921-969, 2009

  The Mathematical Society of Japan

Keywords

• root system
• representation
• tensor product
• multiplicity
• coadjoint orbit
• symplectic quotient

Details

• CRID: 1390001205116604416
• NII Article ID: 10026998507
• NII Book ID: AA0070177X
• DOI: 10.2969/jmsj/06130921
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• NDL BIB ID: 10286995
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I10286995
• Text Lang: en
• Data Source:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
• Abstract License Flag: Disallowed