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

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

DOI: https://doi.org/10.1090/S0002-9947-00-02588-5

Published electronically: April 21, 2000

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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.

References

Susumu Ariki and Kazuhiko Koike, A Hecke algebra of $(\textbf {Z}/r\textbf {Z})\wr {\mathfrak {S}}_n$ and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216–243. MR 1279219, DOI 10.1006/aima.1994.1057
Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18, Springer-Verlag, Berlin, 1989. MR 1002568, DOI 10.1007/978-3-642-74341-2
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
C. W. Curtis, N. Iwahori, and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with $(B,$ $N)$-pairs, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 81–116. MR 347996, DOI 10.1007/BF02684695
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
Charles F. Dunkl, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J. 25 (1976), no. 4, 335–358. MR 407346, DOI 10.1512/iumj.1976.25.25030
Charles F. Dunkl and Donald E. Ramirez, Krawtchouk polynomials and the symmetrization of hypergroups, SIAM J. Math. Anal. 5 (1974), 351–366. MR 346213, DOI 10.1137/0505039
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
V. A. Groza and I. I. Kachurik, Addition and multiplication theorems for Krawtchouk, Hahn and Racah $q$-polynomials, Dokl. Akad. Nauk Ukrain. SSR Ser. A 5 (1990), 3–6, 89 (Russian, with English summary). MR 1071402
P.N. Hoefsmit, Representations of Hecke Algebras of Finite Groups with BN-pairs of Classical Type, thesis, Univ. British Columbia, Vancouver, 1974.
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Michio Jimbo, A $q$-analogue of $U({\mathfrak {g}}{\mathfrak {l}}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252. MR 841713, DOI 10.1007/BF00400222
Tom H. Koornwinder, Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal. 13 (1982), no. 6, 1011–1023. MR 674770, DOI 10.1137/0513072
Tom H. Koornwinder, Askey-Wilson polynomials as zonal spherical functions on the $\textrm {SU}(2)$ quantum group, SIAM J. Math. Anal. 24 (1993), no. 3, 795–813. MR 1215439, DOI 10.1137/0524049
I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161–174. MR 322069, DOI 10.1007/BF01431421
I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), Exp. No. 797, 4, 189–207. Séminaire Bourbaki, Vol. 1994/95. MR 1423624
Hideya Matsumoto, Analyse harmonique dans les systèmes de Tits bornologiques de type affine, Lecture Notes in Mathematics, Vol. 590, Springer-Verlag, Berlin-New York, 1977 (French). MR 0579177, DOI 10.1007/BFb0086707
E. M. Opdam, A remark on the irreducible characters and fake degrees of finite real reflection groups, Invent. Math. 120 (1995), no. 3, 447–454. MR 1334480, DOI 10.1007/BF01241138
Dennis Stanton, Some $q$-Krawtchouk polynomials on Chevalley groups, Amer. J. Math. 102 (1980), no. 4, 625–662. MR 584464, DOI 10.2307/2374091
Dennis Stanton, Three addition theorems for some $q$-Krawtchouk polynomials, Geom. Dedicata 10 (1981), no. 1-4, 403–425. MR 608153, DOI 10.1007/BF01447435
Dennis Stanton, Orthogonal polynomials and Chevalley groups, Special functions: group theoretical aspects and applications, Math. Appl., Reidel, Dordrecht, 1984, pp. 87–128. MR 774056