How to sharpen a tridiagonal pair

Abstract

Let denote a field and let V denote a vector space over 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 Vii=0d of the eigenspaces of A such that A* Vi ⊂eq Vi-1 + Vi + Vi+1 for 0 ≤ i ≤ d, where V-1=0 and Vd+1=0; (iii) there exists an ordering V*ii=0δ of the eigenspaces of A* such that A V*i ⊂eq 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 ⊂eq W, A* W ⊂eq 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 Vi, V*i, Vd-i, V*d-i coincide. Denote this common dimension by i and call A,A* sharp whenever 0=1. Let T denote the -subalgebra of End(V) generated by A,A*. We show: (i) the center Z(T) is a field whose dimension over is 0; (ii) the field Z(T) is isomorphic to each of E0TE0, EdTEd, E*0TE*0, E*dTE*d, where Ei (resp. E*i) is the primitive idempotent of A (resp. A*) associated with Vi (resp. V*i); (iii) with respect to the Z(T)-vector space V the pair A,A* is a sharp tridiagonal pair.