Alpha-determinant cyclic modules and Jacobi polynomials
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- by Kazufumi Kimoto, Sho Matsumoto and Masato Wakayama; with an appendix by Kazufumi Kimoto PDF
- Trans. Amer. Math. Soc. 361 (2009), 6447-6473 Request permission
Abstract:
For positive integers $n$ and $l$, we study the cyclic $\mathcal {U}(\mathfrak {gl}_n)$-module generated by the $l$-th power of the $\alpha$-determinant $\det ^{(\alpha )}(X)$. This cyclic module is isomorphic to the $n$-th tensor space $\mathcal {S}^l(\mathbb {C}^n)^{\otimes n}$ of the symmetric $l$-th tensor space of $\mathbb {C}^n$ for all but finitely many exceptional values of $\alpha$. If $\alpha$ is exceptional, then the cyclic module is equivalent to a proper submodule of $\mathcal {S}^l(\mathbb {C}^n)^{\otimes n}$, i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in $\mathcal {S}^l(\mathbb {C}^n)^{\otimes n}$. 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 $\alpha$ 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 $(\mathfrak {S}_{nl},\mathfrak {S}_l^n)$ 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 $(\mathfrak {S}_{2l},\mathfrak {S}_l^2)$.
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Additional Information
- Kazufumi Kimoto
- Affiliation: Department of Mathematical Sciences, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan
- Email: kimoto@math.u-ryukyu.ac.jp
- Sho Matsumoto
- Affiliation: Faculty of Mathematics, Kyushu University, Hakozaki Higashi-ku, Fukuoka 812-8581, Japan
- Address at time of publication: Department of Mathematics, Nagoya University, Chikusa, Nagoya 464-8602, Japan
- Email: sho-matsumoto@math.nagoya-u.ac.jp
- Masato Wakayama
- Affiliation: Faculty of Mathematics, Kyushu University, Hakozaki Higashi-ku, Fukuoka 812-8581, Japan
- Email: wakayama@math.kyushu-u.ac.jp
- Received by editor(s): October 29, 2007
- Published electronically: July 14, 2009
- Additional Notes: The second author was partially supported by Grant-in-Aid for JSPS Fellows No. 17006193.
The third author was partially supported by Grant-in-Aid for Exploratory Research No. 18654005. - © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 6447-6473
- MSC (2000): Primary 22E47, 33C45; Secondary 43A90, 13A50
- DOI: https://doi.org/10.1090/S0002-9947-09-04860-0
- MathSciNet review: 2538600
Dedicated: Dedicated to Professor Masaaki Yoshida on his sixtieth birthday.