Tridiagonal pairs of type III with height one

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Xue Li
Bo Hou
Suogang Gao


Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension, and let A, A denote a tridiagonal pair on of diameter d.  Let V0, . . . , Vd  denote a standard ordering of  the eigenspaces of A on V , and let θ0, . . . , θddenote the corresponding eigenvalues of A. It is assumed that d ≥ 3.  Let ρi  denote the dimension of Vi. The sequence ρ0, ρ1, . . . , ρdis called the shape of the tridiagonal pair. It is known that ρ0 = 1 and there  exists  a  unique  integer  h (0 ≤ h ≤ d/2)  such  that  ρi−1< ρi  for  1 ≤ i ≤ h,  ρi−1 = ρi  for  h < i ≤ d − h,  and  ρi−1> ρifor d − h < i ≤ d. The integer h is known as the height of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either 1, 2, 2, . . ., 2, 1 or 1, 3, 3, 1.  In each case, an interesting basis is found for V and the actions of A, A on this basis are described.

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