A refinement of the split decomposition of a tridiagonal pair Original Research Article

Let V denote a nonzero finite dimensional vector space over a field image, and let (A, A*) denote a tridiagonal pair on V of diameter d. Let V = U0 + cdots, three dots, centered + Ud denote the split decomposition, and let ρi denote the dimension of Ui. In this paper, at first we show there exists a unique integer h (0 less-than-or-equals, slant h less-than-or-equals, slant d/2) such that ρi−1 < ρi for 1 less-than-or-equals, slant i less-than-or-equals, slant h, ρi−1 = ρi for h < i less-than-or-equals, slant d − h and ρi−1 > ρi for d − h < i less-than-or-equals, slant d. We call h the height of the tridiagonal pair. For 0 less-than-or-equals, slant r less-than-or-equals, slant h, we define subspaces image (r less-than-or-equals, slant i less-than-or-equals, slant d − r) by image, where R denotes the rasing map. We show V is decomposed as a direct sum image. This gives a refinement of the split decomposition. Define image, and observe image. We show LU(r)subset of or equal toU(r-1)+U(r)+U(r+1) for 0 less-than-or-equals, slant r less-than-or-equals, slant h, where we set U(−1) = U(h+1) = 0. Let F(r):V→U(r) denote the projection. We show the lowering map L is decomposed as L = L(−) + L(0) + L(+), where image, image, and image. These maps satisfy L(-)U(r)subset ofU(r-1),L(0)U(r)subset of or equal toU(r), and L(+)U(r)subset of or equal toU(r+1) for 0 less-than-or-equals, slant r less-than-or-equals, slant h. The main results of this paper are the following: (i) For 0 less-than-or-equals, slant r less-than-or-equals, slant h − 1 and r + 2 less-than-or-equals, slant i less-than-or-equals, slant d − r − 1, RL(+) = αL(+)R holds on image for some scalar α; (ii) For 1 less-than-or-equals, slant r less-than-or-equals, slant h and r less-than-or-equals, slant i less-than-or-equals, slant d − r − 1, RL(−) = βL(−)R holds on image for some scalar β; (iii) For 0 less-than-or-equals, slant r less-than-or-equals, slant h and r + 1 less-than-or-equals, slant i less-than-or-equals, slant d − r − 1, RL(0) = βL(0)R + γI holds on image for some scalars γ, δ. Moreover we give explicit expressions of α, β, γ, δ.