A classification of sharp tridiagonal pairs

AbstractLet 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 dimV0=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