Sharp tridiagonal pairs

Abstract

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of K -linear transformations A : V → V and A ∗ : V → V that satisfies 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 . A conjecture of Tatsuro Ito and the second author states that if K is algebraically closed then A , A ∗ is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A , A ∗ is sharp and using the data Φ = ( A ; { V i } i = 0 d ; A ∗ ; { V i ∗ } i = 0 d ) we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by ( A ∗ ; { V i ∗ } i = 0 d ; A ; { V i } i = 0 d ) or ( A ; { V d - i } i = 0 d ; A ∗ ; { V i ∗ } i = 0 d ) or ( A ; { V i } i = 0 d ; A ∗ ; { V d - i ∗ } i = 0 d ) . We prove that if the isomorphism class of Φ is determined by the parameter array then there exists a nondegenerate symmetric bilinear form 〈 , 〉 on V such that 〈 Au , v 〉 = 〈 u , Av 〉 and 〈 A ∗ u , v 〉 = 〈 u , A ∗ v 〉 for all u , v ∈ V .