Tridiagonal pairs and the [formula omitted]-conjecture

Abstract

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 { V i } i = 0 d of the eigenspaces of A such that A ∗ V i ⊆ V i - 1 + V i + V i + 1 for 0 ⩽ i ⩽ d , where V - 1 = 0 and V d + 1 = 0 ; (iii) there exists an ordering { V i ∗ } i = 0 δ of the eigenspaces of A ∗ such that AV i ∗ ⊆ V i - 1 ∗ + V i ∗ + V i + 1 ∗ for 0 ⩽ i ⩽ δ , where V - 1 ∗ = 0 and V δ + 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 V i , V d - i , V i ∗ , V d - i ∗ coincide. We say the pair A , A ∗ is sharp whenever dim V 0 = 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 { θ i } i = 0 d , { θ i ∗ } i = 0 d 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 e 0 ∗ Te 0 ∗ for a certain idempotent e 0 ∗ ∈ T . Let F [ x 1 , … , x d ] denote the polynomial algebra over F involving d mutually commuting indeterminates. We display a surjective F -algebra homomorphism μ : F [ x 1 , … , x d ] → e 0 ∗ Te 0 ∗ . We conjecture that μ is an isomorphism. We show that this μ -conjecture implies the classification conjecture, and that the μ -conjecture holds for d ⩽ 5 .