Alpha-determinant cyclic modules and jacobi polynomials

Kazufumi Kimoto, Sho Matsumoto, Masato Wakayama, Kimoto Kazufumi Kimoto

Research output: Contribution to journalArticle

Abstract

For positive integers n and l, we study the cyclic U(gln)-module generated by the l-th power of the α -determinant det (α)(X). This cyclic module is isomorphic to the n-th tensor space Sl(ℂn)× of the symmetric l-th tensor space of Cn for all but finitely many exceptional values of α. If α is exceptional, then the cyclic module is equivalent to a proper submodule of Sl(ℂn)×, i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in Sl(ℂn)×. The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and whose entries are polynomials in a with rational coefficients. In particular, we determine the matrix completely when n = 2. In this case, the matrix becomes a scalar and is essentially given by a classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary. In the Appendix, we consider a variation of the spherical Fourier transformation for (Cnl,Cnl ) as a main tool for analyzing the same problems, and describe the case where n = 2 by using the zonal spherical functions of the Gelfand pair (C 2l,C2l ). copy; 2009 American Mathematical Society.

Original languageEnglish
Pages (from-to)6447-6473
Number of pages27
JournalTransactions of the American Mathematical Society
Volume361
Issue number12
DOIs
Publication statusPublished - Dec 1 2009

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Jacobi Polynomials
Determinant
Polynomials
Module
Tensors
Tensor
Gelfand Pairs
Spherical Functions
Polynomial
Fourier Transformation
Degeneration
Multiplicity
Isomorphic
Scalar
Integer
Coefficient

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

Alpha-determinant cyclic modules and jacobi polynomials. / Kimoto, Kazufumi; Matsumoto, Sho; Wakayama, Masato; Kazufumi Kimoto, Kimoto.

In: Transactions of the American Mathematical Society, Vol. 361, No. 12, 01.12.2009, p. 6447-6473.

Research output: Contribution to journalArticle

Kimoto, Kazufumi ; Matsumoto, Sho ; Wakayama, Masato ; Kazufumi Kimoto, Kimoto. / Alpha-determinant cyclic modules and jacobi polynomials. In: Transactions of the American Mathematical Society. 2009 ; Vol. 361, No. 12. pp. 6447-6473.
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