# Tridiagonal pairs of type III with height one

## Main Article Content

## Abstract

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 *V *of diameter *d*. Let *V*_{0}*, . . . , V _{d} *denote a standard ordering of the eigenspaces of

*A*on

*V*, and let

*θ*

_{0}

*, . . . , θ*denote the corresponding eigenvalues of

_{d}*A*. It is assumed that

*d ≥*3. Let

*ρ*denote the dimension of

_{i}*V*. The sequence

_{i}*ρ*

_{0}

*, ρ*

_{1}

*, . . . , ρ*is called the

_{d}*shape*of the tridiagonal pair. It is known that

*ρ*

_{0}= 1 and there exists a unique integer

*h*(0

*≤ h ≤ d/*2) such that

*ρ*

_{i}_{−1}

*< ρ*for 1

_{i}*≤ i ≤ h*,

*ρ*

_{i}_{−1}=

*ρ*for

_{i}*h < i ≤ d − h*, and

*ρ*

_{i}_{−1}

*> ρ*for

_{i}*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.