Tridiagonal pairs and the μ-conjecture

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 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 of the eigenspaces of A∗ such that for 0⩽i⩽δ, where and ; (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 coincide. We say the pair A,A∗ is sharp whenever dimV0=1. It is known that if F is algebraically closed then A,A∗ is sharp. A conjectured classification of the sharp tridiagonal pairs was recently introduced by T. Ito and the second author. We present a result which supports the conjecture. Given scalars in F that satisfy the known constraints on the eigenvalues of a tridiagonal pair, we define an F-algebra T by generators and relations. We consider the F-algebra for a certain idempotent . Let F[x1,…,xd] denote the polynomial algebra over F involving d mutually commuting indeterminates. We display a surjective F-algebra homomorphism . We conjecture thatμ is an isomorphism. We show that this μ-conjecture implies the classification conjecture, and that the μ-conjecture holds for d⩽5.