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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Alpha-determinant cyclic modules and Jacobi polynomials


Authors: Kazufumi Kimoto, Sho Matsumoto and Masato Wakayama; with an appendix by Kazufumi Kimoto
Journal: Trans. Amer. Math. Soc. 361 (2009), 6447-6473
MSC (2000): Primary 22E47, 33C45; Secondary 43A90, 13A50
Published electronically: July 14, 2009
MathSciNet review: 2538600
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04860-0
PII: S 0002-9947(09)04860-0
Keywords: Alpha-determinant, cyclic modules, Jacobi polynomials, singly confluent Heun ODE, permanent, Kostka numbers, irreducible decomposition, spherical Fourier transformation, zonal spherical functions, Gelfand pair
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.
Dedicated: Dedicated to Professor Masaaki Yoshida on his sixtieth birthday.
Article copyright: © Copyright 2009 American Mathematical Society