×

zbMATH — the first resource for mathematics

\(q\)-Krawtchouk polynomials as spherical functions on the Hecke algebra of type \(B\). (English) Zbl 0957.33014
The author unifies several interpretations of Krawtchouk and \(q\)-Krawtchouk polynomials by deriving them as zonal spherical functions for the Hecke algebra for the hyperoctahedral group. Let \({\mathcal H}_n\) be the Hecke algebra for the hyperoctahedral group and \({\mathcal F}_n\) the subalgebra corresponding to the Hecke algebra for the symmetric group. Consider the representation of \({\mathcal H}_n\) induced from the index representation of \({\mathcal F}_n\), and let \(V_n\) be the induced module. \(V_n\) is a commutative algebra carrying a non-degenerate bilinear form. The \({\mathcal F}_n\)-invariant elements of \(V_n\) can be identified with an irreducible module of \(U_{q^{1/2}}(\mathfrak{sl}(2,\mathbb{C}))\), the quantized universal enveloping algebra for \(\mathfrak{sl}(2,\mathbb{C})\) acting on \(V_n\). The author calculates the characters of \(V_n\) and gives an explicit orthogonal basis for the \({\mathcal F}_n\)-invariant elements of \(V_n\) as eigenvectors of a particular selfadjoint operator. The author finds the second order \(q\)-difference equation satisfied by the corresponding zonal spherical functions and uses it to connect these to previously studied \(q\)-Krawtchouk polynomials.

MSC:
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
20C08 Hecke algebras and their representations
43A90 Harmonic analysis and spherical functions
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Susumu Ariki and Kazuhiko Koike, A Hecke algebra of (\?/\?\?)\?\?_\? and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216 – 243. · Zbl 0840.20007 · doi:10.1006/aima.1994.1057 · doi.org
[2] 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]. · Zbl 1107.13001
[3] 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. · Zbl 0747.05073
[4] 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.
[5] Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1995. Corrected reprint of the 1994 original. · Zbl 0839.17009
[6] C. W. Curtis, N. Iwahori, and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with (\?, \?)-pairs, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 81 – 116. · Zbl 0254.20004
[7] Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. · Zbl 0469.20001
[8] 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. · Zbl 0616.20001
[9] Charles F. Dunkl, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J. 25 (1976), no. 4, 335 – 358. · Zbl 0326.33008 · doi:10.1512/iumj.1976.25.25030 · doi.org
[10] Charles F. Dunkl and Donald E. Ramirez, Krawtchouk polynomials and the symmetrization of hypergroups, SIAM J. Math. Anal. 5 (1974), 351 – 366. · Zbl 0249.43006 · doi:10.1137/0505039 · doi.org
[11] 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. · Zbl 0695.33001
[12] V. A. Groza and I. I. Kachurik, Addition and multiplication theorems for Krawtchouk, Hahn and Racah \?-polynomials, Dokl. Akad. Nauk Ukrain. SSR Ser. A 5 (1990), 3 – 6, 89 (Russian, with English summary). · Zbl 0725.33013
[13] P.N. Hoefsmit, Representations of Hecke Algebras of Finite Groups with BN-pairs of Classical Type, thesis, Univ. British Columbia, Vancouver, 1974.
[14] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028
[15] Michio Jimbo, A \?-analogue of \?(\?\?(\?+1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247 – 252. · Zbl 0602.17005 · doi:10.1007/BF00400222 · doi.org
[16] Tom H. Koornwinder, Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal. 13 (1982), no. 6, 1011 – 1023. · Zbl 0505.33015 · doi:10.1137/0513072 · doi.org
[17] Tom H. Koornwinder, Askey-Wilson polynomials as zonal spherical functions on the \?\?(2) quantum group, SIAM J. Math. Anal. 24 (1993), no. 3, 795 – 813. · Zbl 0799.33015 · doi:10.1137/0524049 · doi.org
[18] I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161 – 174. · Zbl 0286.20062 · doi:10.1007/BF01431421 · doi.org
[19] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), Exp. No. 797, 4, 189 – 207. Séminaire Bourbaki, Vol. 1994/95.
[20] 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). · Zbl 0366.22001
[21] 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. · Zbl 0824.20038 · doi:10.1007/BF01241138 · doi.org
[22] Dennis Stanton, Some \?-Krawtchouk polynomials on Chevalley groups, Amer. J. Math. 102 (1980), no. 4, 625 – 662. · Zbl 0448.33019 · doi:10.2307/2374091 · doi.org
[23] Dennis Stanton, Three addition theorems for some \?-Krawtchouk polynomials, Geom. Dedicata 10 (1981), no. 1-4, 403 – 425. · Zbl 0497.43006 · doi:10.1007/BF01447435 · doi.org
[24] Dennis Stanton, Orthogonal polynomials and Chevalley groups, Special functions: group theoretical aspects and applications, Math. Appl., Reidel, Dordrecht, 1984, pp. 87 – 128. · Zbl 0578.20041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.