A refinement of the split decomposition of a tridiagonal pair

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Let V denote a nonzero finite dimensional vector space over a field K , and let ( A, A ∗) denote a tridiagonal pair on V of diameter d. Let V = U 0 + ⋯ + U d denote the split decomposition, and let ρ i denote the dimension of U i . In this paper, at first we show there exists a unique integer h (0 ⩽ h ⩽ d/2) such that ρ i−1 < ρ i for 1 ⩽ i ⩽ h, ρ i−1 = ρ i for h < i ⩽ d − h and ρ i−1 > ρ i for d − h < i ⩽ d. We call h the height of the tridiagonal pair. For 0 ⩽ r ⩽ h, we define subspaces U i ( r ) ( r ⩽ i ⩽ d − r) by U i ( r ) = R i - r ( U r ∩ Ker R d - 2 r + 1 ) , where R denotes the rasing map. We show V is decomposed as a direct sum V = ∑ r = 0 h ∑ i = r d - r U i ( r ) . This gives a refinement of the split decomposition. Define U ( r ) = ∑ i = r d - r U i ( r ) , and observe V = ∑ r = 0 h U ( r ) . We show LU ( r ) ⊆ U ( r - 1 ) + U ( r ) + U ( r + 1 ) for 0 ⩽ r ⩽ 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 L ( - ) = ∑ r = 1 h F ( r - 1 ) LF ( r ) , L ( 0 ) = ∑ r = 0 h F ( r ) LF ( r ) , and L ( + ) = ∑ r = 0 h - 1 F ( r + 1 ) LF ( r ) . These maps satisfy L ( - ) U ( r ) ⊂ U ( r - 1 ) , L ( 0 ) U ( r ) ⊆ U ( r ) , and L ( + ) U ( r ) ⊆ U ( r + 1 ) for 0 ⩽ r ⩽ h. The main results of this paper are the following: (i) For 0 ⩽ r ⩽ h − 1 and r + 2 ⩽ i ⩽ d − r − 1, RL (+) = αL (+) R holds on U i ( r ) for some scalar α; (ii) For 1 ⩽ r ⩽ h and r ⩽ i ⩽ d − r − 1, RL (−) = βL (−) R holds on U i ( r ) for some scalar β; (iii) For 0 ⩽ r ⩽ h and r + 1 ⩽ i ⩽ d − r − 1, RL (0) = βL (0) R + γI holds on U i ( r ) for some scalars γ, δ. Moreover we give explicit expressions of α, β, γ, δ.