Abstract
Let F denote a field and let V denote a vector space over F with finite positive dimension. Consider a pair A, A* of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of V that swaps A and A*. Such an automorphism is unique, and called the duality A ↔ A*. In the present paper we give a comprehensive description of this duality. In particular,we display an invertible F-linearmap T on V such that the map X → TXT−1is the duality A ↔ A*. We express T as a polynomial in A and A*. We describe how T acts on 4 flags, 12 decompositions, and 24 bases for V.
References
[1] B. Curtin, Distance-regular graps which supports a spin model are thin, Discrete Math. 197/198 (1999), 205-216.10.1016/S0012-365X(99)90065-1Search in Google Scholar
[2] B. Curtin, Modular Leonard triples, Linear algebra Appl. 424 (2007), 510-539.10.1016/j.laa.2007.02.024Search in Google Scholar
[3] B. Curtin, K. Nomura, Spin models and strongly hyper-self-dual Bose-Mesner algebras, J. Algebraic Combin. 13 (2004), 173-186.10.1023/A:1011297515395Search in Google Scholar
[4] J.T. Go, The Terwilliger algebra of the hypercube, European J. Comb. 23 (2002), 399-429.10.1006/eujc.2000.0514Search in Google Scholar
[5] T. Ito, H. Rosengren, P. Terwilliger, Evaluation modules for the q-tetrahedron algebra, Linear Algebra Appl. 451 (2014), 107-168.10.1016/j.laa.2014.03.019Search in Google Scholar
[6] K. Nomura, P. Terwilliger, Matrix units associated with the split basis of a Leonard pair, Linear Algebra Appl. 418 (2006), 775-787.10.1016/j.laa.2006.03.009Search in Google Scholar
[7] K. Nomura, P. Terwilliger, Affine transformations of a Leonard pair, Electron. J. Linear Algebra 16 (2007), 389-418.10.13001/1081-3810.1210Search in Google Scholar
[8] K. Nomura, P. Terwilliger, The switching element for a Leonard pair, Linear Algebra Appl. 428 (2008), 1083-1108.10.1016/j.laa.2007.09.002Search in Google Scholar
[9] K.Nomura, P. Terwilliger, Transitionmaps between the 24 bases for a Leonard pair, Linear Algebra Appl. 431 (2009), 571-593.10.1016/j.laa.2009.03.007Search in Google Scholar
[10] K. Nomura, P. Terwilliger, Krawtchouk polynomials, the Lie algebra sl2, and Leonard pairs, Linear Algebra Appl. 437 (2012), 345-375.10.1016/j.laa.2012.02.006Search in Google Scholar
[11] K. Nomura, P. Terwilliger, Totally bipartite tridiagonal pairs, preprint, arXiv:1711.00332.Search in Google Scholar
[12] J. Rotman, Advanced Modern Algebra, 2nd edition, AMS, Providence, RI, 2010.10.1090/gsm/114Search in Google Scholar
[13] P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), 149-203.10.1016/S0024-3795(01)00242-7Search in Google Scholar
[14] P. Terwilliger, Leonard pairs from 24 points of view, Rocky Mountain J. Math. 32 (2002), 827-887.10.1216/rmjm/1030539699Search in Google Scholar
[15] P. Terwilliger, Leonard pairs and the q-Racah polynomials, Linear Algebra Appl. 387 (2004), 235-276.10.1016/j.laa.2004.02.014Search in Google Scholar
[16] P. Terwilliger, An algebraic approach to the Askey scheme of orthogonal polynomials, Orthogonal polynomials and special functions, Lecture Notes in Math., 1883, Springer, Berlin, 2006, pp. 255-330, arXiv:math/0408390.10.1007/978-3-540-36716-1_6Search in Google Scholar
© by Kazumasa Nomura, Paul Terwilliger, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
