Towards a classification of the tridiagonal pairs

Let K denote a field and let V denote a vector space over K with finite positive dimension. Let End(V ) denote the K-algebra consisting of all K-linear transformations from V to V . We consider a pair A,A ∈ End(V ) that satisfy (i)–(iv) below: (i) Each of A,A is diagonalizable. (ii) There exists an ordering {Vi} d i=0 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 {V ∗ i } δ i=0 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 , AW ⊆ W , W 6= 0, W 6= V . We call such a pair a tridiagonal pair on V . Let E 0 denote the element of End(V ) such that (E 0 − I)V ∗ 0 = 0 and E ∗ 0V ∗ i = 0 for 1 ≤ i ≤ d. Let D (resp. D ) denote the K-subalgebra of End(V ) generated by A (resp. A). In this paper we prove that the span of E 0DD DE 0 equals the span of E ∗ 0DE ∗ 0DE ∗ 0 , and that the elements of E ∗ 0DE ∗ 0 mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs.