The traces associated with a sharp tridiagonal system

Abstract

Let K denote a field and let V denote a vector space over K with finite positive dimension. Let 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⁎) and for 0≤i≤d let Vi (resp. Vi⁎) be the eigenspace of A (resp. A⁎) associated with θi (resp. θi⁎). It is known that Vi,Vi⁎ have the same dimension. The tridiagonal pair A,A⁎ is said to be sharp whenever dim(V0)=1. For 0≤i≤d, let Ei (resp. Ei⁎) denote the primitive idempotent of A (resp. A⁎) associated with θi (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⁎ 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≤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.