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 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. Let denote the element of End(V) such that and 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 equals the span of , and that the elements of 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.