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

Koelink, H.

doi:10.1090/S0002-9947-00-02588-5

$q$-Krawtchouk polynomials as spherical functions on the Hecke algebra of type $B$
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by H. T. Koelink PDF

Trans. Amer. Math. Soc. 352 (2000), 4789-4813 Request permission

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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