Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): H. T. Koelink
Journal: Trans. Amer. Math. Soc. 352 (2000), 4789-4813.
MSC (2000): Primary 33D80, 20C08, 43A90
Posted: April 21, 2000
MathSciNet review: 1707197
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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 -Krawtchouk polynomials using the quantised enveloping algebra for ${\mathfrak{sl}}(2,\mathbb{C} )$ . The result covers a number of previously established interpretations of (-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum group.

References:

1.: S. Ariki and K. Koike, A Hecke algebra of $(\mathbb{Z} /r\mathbb{Z} )\wr {\mathfrak{S}}_{n}$ and construction of its irreducible representations, Adv. Math. 106 (1994), 216-243. MR 95h:20006
2.: N. Bourbaki, Groupes et Algèbres de Lie, Chapitres 4, 5 and 6, Masson, 1981. MR 83g:17001
3.: A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-Regular Graphs, Ergebnisse Math. 3/18, Springer-Verlag, 1989. MR 90e:05001
4.: R.W. Carter, Finite Groups of Lie Type, Wiley-Interscience, 1985. MR 94k:20020
5.: V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, 1994. MR 95j:17010; corrected reprint MR 96h:17014
6.: C.W. Curtis, N. Iwahori and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with -pairs, Publ. Math. IHES 40 (1971), 81-116. MR 50:494
7.: C.W. Curtis and I. Reiner, Methods of Representation Theory, vol. 1, Wiley-Interscience, 1981. MR 82i:20001
8.: -, Methods of Representation Theory, vol. 2, Wiley-Interscience, 1987. MR 88f:20002
9.: C.F. Dunkl, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J. 25 (1976), 335-358. MR 53:11122
10.: C.F. Dunkl and D.E. Ramirez, Krawtchouk polynomials and the symmetrization of hypergroups, SIAM J. Math. Anal. 5 (1974), 351-366. MR 49:10939
11.: G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, 1990. MR 91d:33034
12.: V.A. Groza and I.I. Kachurik, Addition and multiplication theorems for Krawtchouk, Hahn and Racah -polyomials, Doklady Akad. Nauk Ukraine SSR, Ser. A 89 (1990), 3-6, (in Russian). MR 91m:33031
13.: P.N. Hoefsmit, Representations of Hecke Algebras of Finite Groups with BN-pairs of Classical Type, thesis, Univ. British Columbia, Vancouver, 1974.
14.: J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univ. Press, 1990. MR 92h:20002
15.: M. Jimbo, A -analogue of $U({\mathfrak{gl}}(N+1))$ , Hecke algebra and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252. MR 87k:17011
16.: T.H. Koornwinder, Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal. 13 (1982), 1011-1023. MR 84f:33022
17.: -, Askey-Wilson polynomials as zonal spherical functions on the quantum group, SIAM J. Math. Anal. 24 (1993), 795-813. MR 94k:33042
18.: I.G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161-174. MR 48:433
19.: -, Affine Hecke algebras and orthogonal polynomials, Sém. Bourbaki 1994/5, 797, Astérisque 237 (1996), 189-207. MR 99f:33024
20.: H. Matsumoto, Analyse Harmonique dans les Systèmes de Tits Bornologiques de Type Affine, LNM 590, Springer-Verlag, 1977. MR 58:28315
21.: E.M. Opdam, A remark on the irreducible characters and fake degrees of finite real reflection groups, Invent. Math. 120 (1995), 447-454. MR 96e:20011
22.: D. Stanton, Some -Krawtchouk polynomials on Chevalley groups, Amer. J. Math. 102 (1980), 625-662. MR 82a:33015
23.: -, Three addition theorems for some -Krawtchouk polynomials, Geom. Ded. 10 (1981), 403-425. MR 82g:33035
24.: -, Orthogonal polynomials and Chevalley groups, Special Functions: Group Theoretical Aspects and Applications (R.A. Askey, T.H. Koornwinder, W. Schempp, eds.), Reidel, 1984, pp. 87-128. MR 86d:22008

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|