Alphadeterminant 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), 64476473
MSC (2000):
Primary 22E47, 33C45; Secondary 43A90, 13A50
Published electronically:
July 14, 2009
MathSciNet review:
2538600
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References 
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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 9030213, Japan
Email:
kimoto@math.uryukyu.ac.jp
Sho Matsumoto
Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki Higashiku, Fukuoka 8128581, Japan
Address at time of publication:
Department of Mathematics, Nagoya University, Chikusa, Nagoya 4648602, Japan
Email:
shomatsumoto@math.nagoyau.ac.jp
Masato Wakayama
Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki Higashiku, Fukuoka 8128581, Japan
Email:
wakayama@math.kyushuu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994709048600
PII:
S 00029947(09)048600
Keywords:
Alphadeterminant,
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 GrantinAid for JSPS Fellows No. 17006193.
The third author was partially supported by GrantinAid for Exploratory Research No. 18654005.
Dedicated:
Dedicated to Professor Masaaki Yoshida on his sixtieth birthday.
Article copyright:
© Copyright 2009
American Mathematical Society
