The structure 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 K-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 {Vi}i=0d 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 {Vi∗}i=0δ of the eigenspaces of A∗ such that AVi∗⊆Vi-1∗+Vi∗+Vi+1∗ for 0⩽i⩽δ, where V-1∗=0 and Vδ+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 Vi,Vd-i,Vi∗,Vd-i∗ coincide.In this paper we show that the following (i)–(iv) hold provided that K is algebraically closed: (i) Each of V0,V0∗,Vd,Vd∗ has dimension 1.(ii) 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.(iii) There exists a unique anti-automorphism of End(V) that fixes each of A,A∗.(iv) The pair A,A∗ is determined up to isomorphism by the data ({θi}i=0d;{θi∗}i=0d;{ζi}i=0d), where θi (resp.θi∗) is the eigenvalue of A (resp.A∗) on Vi (resp.Vi∗), and{ζi}i=0d is the split sequence of A,A∗ corresponding to {θi}i=0d and {θi∗}i=0d