×

zbMATH — the first resource for mathematics

Introduction to Leonard pairs. (English) Zbl 1035.05103
Author’s abstract: In this survey paper we give an elementary introduction to the theory of Leonard pairs. A Leonard pair is defined as follows. Let \(\mathbb{K}\) denote a field and let \(V\) denote a vector space over \(\mathbb{K}\) with finite positive dimension. By a Leonard pair on \(V\) we mean an ordered pair of linear transformations \(A: V\to V\) and \(B: V\to V\) that satisfy conditions (i), (ii) below. (i) There exists a basis for \(V\) with respect to which the matrix representing \(A\) is irreducible tridiagonal and the matrix representing \(B\) is diagonal. (ii) There exists a basis for \(V\) with respect to which the matrix representing \(A\) is diagonal and the matrix representing \(B\) is irreducible tridiagonal.
We give several examples of Leonard pairs and illustrate the use of Leonard pairs in representation theory, combinatorics, and the theory of orthogonal polynomials.

MSC:
05E30 Association schemes, strongly regular graphs
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahn, C.; Shigemoto, K., Onsager algebra and integrable lattice models, Modern phys. lett. A, 6, 38, 3509-3515, (1991) · Zbl 1020.82541
[2] Andrews, G.; Askey, R.; Roy, R., Special functions, (1999), Cambridge University Press Cambridge
[3] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings London · Zbl 0555.05019
[4] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer Berlin · Zbl 0747.05073
[5] Caughman IV, J.S., The Terwilliger algebras of bipartite P- and Q-polynomial schemes, Discrete math., 196, 65-95, (1999) · Zbl 0924.05067
[6] B. Curtin, Distance-regular graphs which support a spin model are thin, in: Proceedings of the 16th British Combinatorial Conference, London, 1997, Discrete Math. 197/198 (1999) 205-216. · Zbl 0929.05095
[7] Curtin, B.; Nomura, K., Distance-regular graphs related to the quantum enveloping algebra of sl(2), J. algebraic combin., 12, 25-36, (2000) · Zbl 0967.05067
[8] Date, E.; Roan, S.S., The structure of quotients of the Onsager algebra by closed ideals, J. phys. A, 33, 3275-3296, (2000) · Zbl 0998.17027
[9] Davies, B., Onsager’s algebra and the dolan – grady condition in the non-self-dual case, J. math. phys., 32, 2945-2950, (1991) · Zbl 0764.17026
[10] L. Dolan, M. Grady, Conserved charges from self-duality, Phys. Rev. D (3) 25 (1982) 1587-1604.
[11] G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge, 1990. · Zbl 0695.33001
[12] Granovskiı̆, Ya.I.; Lutzenko, I.M.; Zhedanov, A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. phys., 217, 1-20, (1992) · Zbl 0875.17002
[13] Granovskiı̆, Ya.A.; Zhedanov, A.S., Nature of the symmetry group of the 6j-symbol, Zh. èksper. teoret. fiz., 94, 49-54, (1988)
[14] Granovskiı̆, Ya.I.; Zhedanov, A.S., Twisted clebsch – gordan coefficients for su_q(2), J. phys. A, 25, L1029-L1032, (1992) · Zbl 0765.17015
[15] Granovskiı̆, Ya.I.; Zhedanov, A.S., Linear covariance algebra for sl_q(2), J. phys. A, 26, L357-L359, (1993) · Zbl 0784.17019
[16] Granovskiı̆, Ya.I.; Zhedanov, A.S., Spherical q-functions, J. phys. A, 26, 4331-4338, (1993) · Zbl 0854.33013
[17] F.A. Grünbaum, Some bispectral musings, in: The Bispectral Problem, Montreal, PQ, 1997, American Mathematical Society Providence, RI, 1998, pp. 31-45.
[18] Grünbaum, F.A.; Haine, L., The q-version of a theorem of Bochner, J. comput. appl. math., 68, 103-114, (1996) · Zbl 0865.33012
[19] Grünbaum, F.A.; Haine, L., Bispectral Darboux transformations: an extension of the krall polynomials, Internat. math. res. notices, 8, 359-392, (1997) · Zbl 1125.37321
[20] Grünbaum, F.A.; Haine, L., Some functions that generalize the askey – wilson polynomials, Comm. math. phys., 184, 173-202, (1997) · Zbl 0871.33009
[21] F.A. Grünbaum, L. Haine, On a q-analogue of the string equation and a generalization of the classical orthogonal polynomials, in: Algebraic Methods and q-Special Functions, Montréal, QC, 1996, American Mathematical Society, Providence, RI, 1999, pp. 171-181.
[22] F.A. Grünbaum, L. Haine, The Wilson bispectral involution: some elementary examples, in: Symmetries and Integrability of Difference Equations, Canterbury, 1996, Cambridge University Press, Cambridge, 1999, pp. 353-369.
[23] Grünbaum, F.A.; Haine, L.; Horozov, E., Some functions that generalize the krall – laguerre polynomials, J. comp. appl. math., 106, 271-297, (1999) · Zbl 0926.33007
[24] T. Ito, K. Tanabe, P. Terwilliger, Some algebra related to P- and Q-polynomial association schemes, in: Codes and Association Schemes, Piscataway, NJ, 1999, American Mathematical Society, Providence, RI, 2001, pp. 167-192. · Zbl 0995.05148
[25] Kassel, C., Quantum groups, (1995), Springer New York · Zbl 0808.17003
[26] R. Koekoek, R.F. Swarttouw, The Askey scheme of hypergeometric orthogonal polyomials and its q-analog, Report 98-17, Delft University of Technology, The Netherlands, 1998, Available at http://aw.twi.tudelft.nl/ koekoek/research.html.
[27] Koelink, H.T., Askey – wilson polynomials and the quantum su(2) groupsurvey and applications, Acta appl. math., 44, 295-352, (1996) · Zbl 0865.33013
[28] Koelink, H.T., q-krawtchouk polynomials as spherical functions on the Hecke algebra of type B, Trans. amer. math. soc., 352, 4789-4813, (2000) · Zbl 0957.33014
[29] Koelink, H.T.; Van der Jeugt, J., Convolutions for orthogonal polynomials from Lie and quantum algebra representations, SIAM J. math. anal., 29, 794-822, (1998) · Zbl 0977.33013
[30] Koelink, H.T.; Van der Jeugt, J., Bilinear generating functions for orthogonal polynomials, Constr. approx., 15, 481-497, (1999) · Zbl 0941.33011
[31] Koornwinder, T.H., Askey – wilson polynomials as zonal spherical functions on the su(2) quantum group, SIAM J. math. anal., 24, 795-813, (1993) · Zbl 0799.33015
[32] Leonard, D., Orthogonal polynomials, duality, and association schemes, SIAM J. math. anal., 13, 656-663, (1982) · Zbl 0495.33006
[33] H. Rosengren, Multivariable Orthogonal Polynomials as Coupling Coefficients for Lie and Quantum Algebra Representations, Centre for Mathematical Sciences, Lund University, Sweden, 1999. · Zbl 0946.33013
[34] Terwilliger, P., The incidence algebra of a uniform poset, Math. appl., 20, 193-212, (1990) · Zbl 0737.05032
[35] Terwilliger, P., The subconstituent algebra of an association scheme, J. algebraic combin., 1, 363-388, (1992) · Zbl 0785.05089
[36] Terwilliger, P., Introduction to leonard pairs and leonard systems, Sūrikaisekikenkyūsho Kōkyūroku, 1109, 67-79, (1999), (Algebraic combinatorics, Kyoto, 1999.) · Zbl 0957.15500
[37] P. Terwilliger, Leonard pairs from 24 points of view, in: Proceedings of the Special Functions 2000, Tempe AZ, 2000, accepted for publication. · Zbl 1040.05030
[38] Terwilliger, P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear algebra appl., 330, 149-203, (2001) · Zbl 0980.05054
[39] P. Terwilliger, Two relations that generalize the q-Serre relations and the Dolan-Grady relations, in: Physics and Combinatorics 1999, Nagoya, World Scientific, River Edge, NJ, 2001, pp. 377-398. · Zbl 1061.16033
[40] Zhedanov, A.S., Hidden symmetry of askey – wilson polynomials, Teoret. mat. fiz., 89, 190-204, (1991) · Zbl 0744.33009
[41] Zhedanov, A.S., Quantum su_q(2) Algebracartesian version and overlaps, Modern phys. lett. A, 7, 1589-1593, (1992) · Zbl 1020.17514
[42] Zhedanov, A.S., Hidden symmetry algebra and overlap coefficients for two ring-shaped potentials, J. phys. A, 26, 4633-4641, (1993)
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.