Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9498538 | Linear Algebra and its Applications | 2005 | 8 Pages |
Abstract
The notion of a tridiagonal pair was introduced by Ito, Tanabe and Terwilliger. Let V denote a nonzero finite dimensional vector space over a field F. A tridiagonal pair on V is a pair (A, A*), where A : V â V and A* : V â V are linear transformations that satisfy some conditions. Assume (A, A*) is a tridiagonal pair on V. Recently Terwilliger and Vidunas showed that if A is multiplicity-free on V, then (A, A*) satisfy the following “Askey-Wilson relation” for some scalars β, γ, γ*, ϱ, ϱ*, Ï, η, η*.A2Aâ-βAAâA+AâA2-γ(AAâ+AâA)-ϱAâ=γâA2+ÏA+ηI,Aâ2A-βAâAAâ+AAâ2-γâ(AâA+AAâ)-ϱâA=γAâ2+ÏAâ+ηâI.In the present paper, we show that, if a tridiagonal pair (A, A*) satisfy the Askey-Wilson relations, then the eigenspaces of A and the eigenspaces of A* have one common dimension, and moreover if F is algebraically closed then that common dimension is 1.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kazumasa Nomura,