Tridiagonal pairs of q-Racah type

JOURNAL OF ALGEBRA

卷:322

Article

Ito, Tatsuro²  Terwilliger, Paul¹ 

[1] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

[2] Kanazawa Univ, Grad Sch Nat Sci & Technol, Div Math & Phys Sci, Kanazawa, Ishikawa 9201192, Japan

关键词: Tridiagonal pair;    Leonard pair;    q-Racah polynomial;   

DOI  :  10.1016/j.jalgebra.2009.04.008

来源: Elsevier

【 摘 要 】

Let F denote an algebraically closed field and let V denote a vector space over IF with finite positive dimension. We consider a pair of linear transformations A : V -> V and A* : V -> V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering {Vi}(i=0)(d) of the eigenspaces of A such that A* V-i superset of Vi-1 + V-i + Vi+1 for 0 <= i <= d, where V-1 = 0 and Vd+1 = 0; (iii) there exists an ordering {V-j*}(j=0)(delta) of the eigenspaces of A* such that AV(j)* subset of V*(i-1) + V-j* + Vi+1* for 0 <= i <= delta, where V-1* = 0 and V delta+1* = 0; (iv) there is no subspace W of V such that AW subset of W, A*W subset of W, W not equal 0, W not equal V. We call such a pair a tridiagonal pair on V. It is known that d = delta. For 0 <= i <= d let theta(i) (resp. theta(i)*) denote the eigenvalue of A (resp. A*) associated with V-i (resp. V-i*). The pair A, A* is said to have q-Racah type whenever theta(i) = a + bq(2i-d) + cq(d-2i) and theta(i)* = a* + b*q(2i-d) + c*q(d-2i) for 0 <= i <= d, where q, a, b, c, a*, b*, c* are scalars in F with q, b, c, b*, c* nonzero and q(2) is not an element of {1, -1}. This type is the most general one. We classify up to isomorphism the tridiagonal pairs over IF that have q-Racah type. Our proof involves the representation theory of the quantum affine algebra U-q((S) over capl(2)). (C) 2009 Elsevier Inc. All rights reserved.

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