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The shape of a tridiagonal pair. (English) Zbl 1037.17013
Let $$V$$ be a vector space over the field $$F$$ with finite positive dimension. By a tridiagonal pair on $$V$$, we mean a pair $$(A,A^*)$$ of linear transformations $$V$$ that satisfy the following conditions:
(i) each of $$A,A^*$$ is diagonalizable,
(ii) there exists an ordering $$V_0,\dots,V_d$$ of the eigenspaces of $$A$$ such that $$A^*V_i\subseteq V_{i-1}+V_i+V_{i+1}$$ ($$0\leq i\leq d$$), where $$V_{-1}=V_{d+1}=0$$,
(iii) there exists an ordering $$V_0^*,...,V_\delta^*$$ of the eigenspaces of $$A^*$$ such that $$AV_i^*\subseteq V_{i-1}^*+V_i^*+V_{i+1}^*$$ ($$0\leq i\leq \delta$$), where $$V_{-1}^*=V_{\delta+1}^*=0$$,
(iv) there does not exist a subspace $$W$$ of $$V$$, $$0\neq W\neq V$$ such that $$AW\subseteq W$$, $$A^*W\subseteq W$$.
It is known that $$d=\delta$$ and the spaces $$V_i,V_i^*$$ have the same dimension $$\rho_i$$. The sequence ($$\rho_0,\dots,\rho_d$$) is symmetric and unimodal and it is called the shape vector of $$A,A^*$$.
Theorem 1.6. Let $$A,A^*$$ be a tridiagonal pair on $$V$$, $$F$$ be an algebraically closed field with characteristic 0. If $$q\in F$$ is a nonzero scalar that is not a root of unity, $$\beta=q^2+q^{-2}$$ and $$[A,A^2A^*-\beta AA^*A+A^*A^2]=0$$, $$[A^*,A^{*2}A-\beta A^*AA^*+AA^{*2}]=0$$ (where $$[r,s]=rs-sr$$), then $$\rho_i\leq {d\choose i}$$ ($$0\leq i\leq d$$).

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 15A21 Canonical forms, reductions, classification 05E35 Orthogonal polynomials (combinatorics) (MSC2000)
##### Keywords:
tridiagonal pair; shape vector
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##### References:
 [1] Andrews, G, The theory of partitions, (1984), Cambridge University Press Cambridge [2] Askey, R; Wilson, J.A, A set of orthogonal polynomials that generalize the racah coefficients or 6−j symbols, SIAM J. math. anal., 10, 1008-1016, (1979) · Zbl 0437.33014 [3] Bannai, E; Ito, T, Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings London · Zbl 0555.05019 [4] Chari, V; Pressley, A, Quantum affine algebras, Comm. math. phys., 142, 261-283, (1991) · Zbl 0739.17004 [5] Conway, J.H; Sloane, N.J.A, Sphere packings, lattices and groups, (1988), Springer New York · Zbl 0634.52002 [6] T. Ito, K. Tanabe, P. Terwilliger, Some algebra related to P- and Q-polynomial association schemes, Codes and Association Schemes (Piscataway NJ, 1999), Vol. 56, Amer. Math. Soc., Providence, RI, 2001, pp. 167-192. · Zbl 0995.05148 [7] Kac, V, Infinite dimensional Lie algebras, (1995), Cambridge University Press Cambridge [8] R. Koekoek, R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analog, Reports of the faculty of Technical Mathematics and Informatics, Vol. 98-17, Delft, The Netherlands, 1998; available at http://aw.twi.tudelft.nl/ koekoek/research.html. [9] Leonard, D, Orthogonal polynomials, duality, and association schemes, SIAM J. math. anal., 13, 656-663, (1982) · Zbl 0495.33006 [10] Lusztig, G, Introduction to quantum groups, Progress in mathematics, Vol. 110, (1993), Birkhauser Boston · Zbl 0788.17010 [11] Terwilliger, P, The subconstituent algebra of an association scheme, J. algebraic combin., 1, 363-388, (1992) · Zbl 0785.05089 [12] 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 [13] P. Terwilliger, Two relations that generalize the q-Serre relations and the Dolan-Grady relations. Physics and Combinatorics 1999 (Nagoya), World Sci. Publishing, River Edge, NJ, 2001, pp. 377-398. · Zbl 1061.16033 [14] Terwilliger, P, Leonard pairs from 24 points of view, Rocky mountain J. math., 32, 827-888, (2002) · Zbl 1040.05030 [15] P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form. J. Algebra, submitted for publication. · Zbl 1079.15005 [16] Terwilliger, P, Introduction to leonard pairs (OPSFA, Rome, 2001), J. comput. appl. math., 153, 463-475, (2003) · Zbl 1035.05103 [17] Terwilliger, P, Introduction to leonard pairs and leonard systems, Sūrikaisekikenkyūsho Kōkyūroku, Algebraic combinatorics (Kyoto, 1999), 1109, 67-79, (1999) · Zbl 0957.15500 [18] P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition (OPSFA7, Copenhagen, 2003), J. Comput. Appl. Math., submitted for publication. · Zbl 1069.15005 [19] P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array, Geometric and Algebraic Combinatorics 2, Oisterwijk, The Netherlands, 2002, submitted for publication. · Zbl 1063.05138 [20] P. Terwilliger. Leonard pairs and the q-Racah polynomials. Linear Algebra Appl., submitted for publication. · Zbl 1075.05090 [21] Wakimoto, Minoru, Infinite-dimensional Lie algebras, Translations of mathematical monographs, Vol. 195, (2001), Amer. Math. Soc Providence, RI · Zbl 0956.17014
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