A classification of sharp tridiagonal pairs

作者：Ito Tatsuro; Nomura Kazumasa; Terwilliger Paul^*

来源：Linear Algebra and Its Applications, 2011, 435(8): 1857-1884.

DOI：10.1016/j.laa.2011.03.032

• Tridiagonal pair
• Leonard pair
• q-Racah polynomials

Let IF denote a 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 {V-i}(i=0)(d) of the eigenspaces of A such thatA*V-i subset 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*(i)}(i=0)(delta) of the eigenspaces of A* such that AV*(i) subset of V*(i) + V*(i) + V*(i+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 and for 0 <= i <= d the dimensions of V-i, Vd-i, V*(i), v*(d-1) coincide. The pair A, A* is called sharp whenever dim V-0 = 1. It is known that if IF 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 mu-conjecture.

• 出版日期2011-10-15