Sharp tridiagonal pairs

Let 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 satisfies the following conditions: (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. It is known that d=δ and for 0⩽i⩽d the dimensions of coincide. We say the pair A,A∗ is sharp whenever dimV0=1. A conjecture of Tatsuro Ito and the second author states that if K is algebraically closed then A,A∗ is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A,A∗ is sharp and using the data we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by or or . We prove that if the isomorphism class of Φ is determined by the parameter array then 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.