The split decomposition of a tridiagonal pair

AbstractLet K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy (i)–(iv) below:(i)Each of A, A∗ is diagonalizable.(ii)There exists an ordering V0,V1,…,Vd of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0⩽i⩽d, where V-1=0, Vd+1=0.(iii)There exists an ordering V0∗,V1∗,…,Vδ∗ of the eigenspaces of A∗ such that AVi∗⊆Vi-1∗+Vi∗+Vi+1∗ for 0⩽i⩽δ, where V-1∗=0, Vδ+1∗=0.(iv)There is no subspace W of V such that both AW⊆W,A∗W⊆W, other than W=0 and W=V.We call such a pair a tridiagonal pair on V. In this note we obtain two results. First, we show that each of A, A∗ is determined up to affine transformation by the Vi and Vi∗. Secondly, we characterize the case in which the Vi and Vi∗ all have dimension one. We prove both results using a certain decomposition of V called the split decomposition