The split decomposition of a tridiagonal pair Original Research Article

Let image denote a field and let V denote a vector space over image with finite positive dimension. We consider a pair of linear transformations A : V → V and A* : V → V that satisfy (i)–(iv) below: (i) Each of A, A* is diagonalizable. (ii) There exists an ordering V0, V1, …, Vd of the eigenspaces of A such that A*Visubset of or equal toVi-1+Vi+Vi+1 for 0 less-than-or-equals, slant i less-than-or-equals, slant d, where V-1 = 0, Vd+1 = 0. (iii) There exists an ordering image of the eigenspaces of A* such that image for 0 less-than-or-equals, slant i less-than-or-equals, slant δ, where image, image. (iv) There is no subspace W of V such that both AWsubset of or equal toW,A*Wsubset of or equal toW, other than W = 0 and W = V. We call such a pair a tridiagonal pair on V. In this note we obtain two results. First, we show that each of A, A* is determined up to affine transformation by the Vi and image. Secondly, we characterize the case in which the Vi and image all have dimension one. We prove both results using a certain decomposition of V called the split decomposition.