### Abstract

For positive integers n and l, we study the cyclic U(gl_{n})-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 S^{l}(ℂn)^{×} of the symmetric l-th tensor space of C^{n} for all but finitely many exceptional values of α. If α is exceptional, then the cyclic module is equivalent to a proper submodule of S^{l}(ℂ^{n})^{×}, i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in S^{l}(ℂ^{n})^{×}. 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 (C_{nl},C^{n}_{l} ) 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},C^{2}_{l} ). copy; 2009 American Mathematical Society.

Original language | English |
---|---|

Pages (from-to) | 6447-6473 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 1 2009 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*361*(12), 6447-6473. https://doi.org/10.1090/S0002-9947-09-04860-0

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

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 361, no. 12, pp. 6447-6473. https://doi.org/10.1090/S0002-9947-09-04860-0

}

TY - JOUR

T1 - Alpha-determinant cyclic modules and jacobi polynomials

AU - Kimoto, Kazufumi

AU - Matsumoto, Sho

AU - Wakayama, Masato

AU - Kazufumi Kimoto, Kimoto

PY - 2009/12/1

Y1 - 2009/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77950661325&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950661325&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-09-04860-0

DO - 10.1090/S0002-9947-09-04860-0

M3 - Article

AN - SCOPUS:77950661325

VL - 361

SP - 6447

EP - 6473

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 12

ER -