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Transactions of the American Mathematical Society

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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)$.

References [Enhancements On Off] (What's this?)

  • 1. G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
  • 2. S. Ariki, J. Matsuzawa and I. Terada, Representation of Weyl groups on zero weight spaces of $ \mathfrak{g}$-modules, Algebraic and Topological Theories (Kinosaki, 1984), 546-568, Kinokuniya, Tokyo, 1986. MR 1102274
  • 3. E. Bannai and T. Ito, Algebraic Combinatorics I, Association Schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. MR 0882540 (87m:05001)
  • 4. W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics 129, Readings in Mathematics, Springer-Verlag, New York, 1991. MR 1153249 (93a:20069)
  • 5. I. M. Gel'fand and M. A. Naĭmark, Unitary representations of the Lorentz group, Acad. Sci. USSR J. Phys. 10 (1946), 93-94; Izvestiya Akad. Nauk SSSR Ser. Mat. 11 (1947), 411-504. MR 0017282 (8:132b)
  • 6. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. A foundation for computer science, Second edition, Addison-Wesley Publishing Company, Reading, MA, 1994. MR 1397498 (97d:68003)
  • 7. K. Kimoto, Generalized content polynomials toward $ \alpha$-determinant cyclic modules, Preprint (2007).
  • 8. K. Kimoto and M. Wakayama, Invariant theory for singular $ \alpha$-determinants, J. Combin. Theory Ser. A 115 (2008), no. 1, 1-31. MR 2378855
  • 9. -, Quantum $ \alpha$-determinant cyclic modules of $ \mathcal{U}_q(\mathfrak{gl}_n)$, J. Algebra 313 (2007), 922-956. MR 2329577 (2008d:17021)
  • 10. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edn., Oxford University Press, 1995. MR 1354144 (96h:05207)
  • 11. S. Matsumoto, Alpha-pfaffian, pfaffian point process and shifted Schur measure, Linear Algebra Appl. 403 (2005), 369-398. MR 2140292 (2006d:15014)
  • 12. S. Matsumoto and M. Wakayama, Alpha-determinant cyclic modules of $ \mathfrak{gl}_n (\mathbb{C})$, J. Lie Theory 16 (2006), 393-405. MR 2197599 (2007a:17011)
  • 13. T. Shirai and Y. Takahashi, Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes, J. Funct. Anal. 205 (2003), 414-463. MR 2018415 (2004m:60104)
  • 14. S. Yu. Slavyanov and W. Lay, Special Functions - A Unified Theory Based on Singularities, Oxford Univ. Press, Oxford, 2000. MR 1858237 (2004e:33003)
  • 15. G. Szegö, Orthogonal Polynomials, 4th edn., American Mathematical Society, 1975. MR 0372517 (51:8724)
  • 16. D. Vere-Jones, A generalization of permanents and determinants, Linear Algebra Appl. 111 (1988), 119-124. MR 0974048 (89j:15014)
  • 17. H. Weyl, The Classical Groups: Their Invariants and Representations, Fifteenth printing, Princeton Landmarks in Mathematics, Princeton University Press, 1997. MR 1488158 (98k:01049)

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Additional Information

Kazufumi Kimoto
Affiliation: Department of Mathematical Sciences, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan

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

Masato Wakayama
Affiliation: Faculty of Mathematics, Kyushu University, Hakozaki Higashi-ku, Fukuoka 812-8581, Japan

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

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