The traces associated with a sharp tridiagonal system

Let KK denote a field and let V   denote a vector space over KK with finite positive dimension. Let A,A⁎A,A⁎ denote a tridiagonal pair on V  . Let {θi}i=0d (resp. {θi⁎}i=0d) denote a standard ordering of the eigenvalues of A   (resp. A⁎A⁎) and for 0≤i≤d0≤i≤d let ViVi (resp. Vi⁎) be the eigenspace of A   (resp. A⁎A⁎) associated with θiθi (resp. θi⁎). It is known that Vi,Vi⁎ have the same dimension. The tridiagonal pair A,A⁎A,A⁎ is said to be sharp whenever dim(V0)=1dim(V0)=1. For 0≤i≤d0≤i≤d, let EiEi (resp. Ei⁎) denote the primitive idempotent of A   (resp. A⁎A⁎) associated with θiθi (resp. θi⁎). Then Φ=(A;E0,E1,⋯,Ed;A⁎;E0⁎,E1⁎,⋯,Ed⁎) is a tridiagonal system on V. We say Φ   is sharp whenever the tridiagonal pair A,A⁎A,A⁎ is sharp. Assume Φ   is sharp and let {ζi}i=0d denote the split sequence of Φ  . The sequence ({θi}i=0d;{θi⁎}i=0d;{ζi}i=0d) is called the parameter array of Φ. Recently in [3] K. Nomura and P. Terwilliger proposed the following problem: Let Φ   denote a sharp tridiagonal system. For 0≤i≤d0≤i≤d find each oftr(EiE0⁎),tr(EiEd⁎),tr(Ei⁎E0),tr(Ei⁎Ed) in terms of the parameter array of Φ. In the present paper we solve this problem.