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

DOI:
https://doi.org/10.1090/S0002-9947-09-04860-0

Published electronically:
July 14, 2009

MathSciNet review:
2538600

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Abstract | References | Similar Articles | Additional Information

Abstract: For positive integers and , we study the cyclic -module generated by the -th power of the -determinant . This cyclic module is isomorphic to the -th tensor space of the symmetric -th tensor space of for all but finitely many exceptional values of . If is exceptional, then the cyclic module is equivalent to a *proper* submodule of , i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in . 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 with rational coefficients. In particular, we determine the matrix completely when . 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 as a main tool for analyzing the same problems, and describe the case where by using the zonal spherical functions of the Gelfand pair .

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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:
https://doi.org/10.1090/S0002-9947-09-04860-0

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