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$q$-Krawtchouk polynomials as spherical functions on the Hecke algebra of type $B$


Author: H. T. Koelink
Journal: Trans. Amer. Math. Soc. 352 (2000), 4789-4813
MSC (2000): Primary 33D80, 20C08, 43A90
DOI: https://doi.org/10.1090/S0002-9947-00-02588-5
Published electronically: April 21, 2000
MathSciNet review: 1707197
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Abstract:

The Hecke algebra for the hyperoctahedral group contains the Hecke algebra for the symmetric group as a subalgebra. Inducing the index representation of the subalgebra gives a Hecke algebra module, which splits multiplicity free. The corresponding zonal spherical functions are calculated in terms of $q$-Krawtchouk polynomials using the quantised enveloping algebra for ${\mathfrak{sl}}(2,\mathbb{C} )$. The result covers a number of previously established interpretations of ($q$-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum $SU(2)$ group.


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Additional Information

H. T. Koelink
Affiliation: Department of Mathematics, Delft University of Technology, ITS-TWI-AW, P.O. Box 5031, 2600 GA Delft, the Netherlands
Email: koelink@twi.tudelft.nl

DOI: https://doi.org/10.1090/S0002-9947-00-02588-5
Keywords: Hecke algebra, $q$-Krawtchouk polynomial, zonal spherical function
Received by editor(s): June 3, 1996
Received by editor(s) in revised form: November 1, 1998
Published electronically: April 21, 2000
Additional Notes: Work done at the University of Amsterdam supported by the Netherlands Organization for Scientific Research (NWO) under project number 610.06.100
Article copyright: © Copyright 2000 American Mathematical Society

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