Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9498277 | Linear Algebra and its Applications | 2005 | 23 Pages |
Abstract
Let V denote a nonzero finite dimensional vector space over a field K, and let (A, Aâ) denote a tridiagonal pair on V of diameter d. Let V = U0 + â¯Â + Ud denote the split decomposition, and let Ïi denote the dimension of Ui. In this paper, at first we show there exists a unique integer h (0 ⩽ h ⩽ d/2) such that Ïiâ1 < Ïi for 1 ⩽ i ⩽ h, Ïiâ1 = Ïi for h < i ⩽ d â h and Ïiâ1 > Ïi for d â h < i ⩽ d. We call h the height of the tridiagonal pair. For 0 ⩽ r ⩽ h, we define subspaces Ui(r) (r ⩽ i ⩽ d â r) by Ui(r)=Ri-r(Urâ©KerRd-2r+1), where R denotes the rasing map. We show V is decomposed as a direct sum V=âr=0hâi=rd-rUi(r). This gives a refinement of the split decomposition. Define U(r)=âi=rd-rUi(r), and observe V=âr=0hU(r). We show LU(r)âU(r-1)+U(r)+U(r+1) for 0 ⩽ r ⩽ h, where we set U(â1) = U(h+1) = 0. Let F(r):VâU(r) denote the projection. We show the lowering map L is decomposed as L = L(â) + L(0) + L(+), where L(-)=âr=1hF(r-1)LF(r), L(0)=âr=0hF(r)LF(r), and L(+)=âr=0h-1F(r+1)LF(r). These maps satisfy L(-)U(r)âU(r-1),L(0)U(r)âU(r), and L(+)U(r)âU(r+1) for 0 ⩽ r ⩽ h. The main results of this paper are the following: (i) For 0 ⩽ r ⩽ h â 1 and r + 2 ⩽ i ⩽ d â r â 1, RL(+) = αL(+)R holds on Ui(r) for some scalar α; (ii) For 1 ⩽ r ⩽ h and r ⩽ i ⩽ d â r â 1, RL(â) = βL(â)R holds on Ui(r) for some scalar β; (iii) For 0 ⩽ r ⩽ h and r + 1 ⩽ i ⩽ d â r â 1, RL(0) = βL(0)R + γI holds on Ui(r) for some scalars γ, δ. Moreover we give explicit expressions of α, β, γ, δ.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kazumasa Nomura,