A classification of sharp tridiagonal pairs

金沢大学理工研究域数物科学系Let F denote a field and let V denote a vector space over F 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 = 0d of the eigenspaces of A such that A* Vi ⊆ Vi - 1 + Vi + Vi + 1 for 0 ≤ i ≤ d, where V- 1 = 0 and Vd + 1 = 0; (iii) there exists an ordering {Vi*}i = 0δ of the eigenspaces of A* such that AVi* ⊆ Vi - 1* + Vi* + Vi + 1* for 0 ≤ i ≤ δ, where V- 1* = 0 and Vδ + 1* = 0; (iv) there is no subspace W of V such that AW ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a tridiagonal pair on V. It is known that d = δ and for 0 ≤ i ≤ d the dimensions of Vi, Vd - i, Vi*, Vd - i* coincide. The pair A, A* is called sharp whenever dim V0 = 1. It is known that if F is algebraically closed then A, A* is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture. © 2011 Elsevier Inc. All rights reserved