Abstract :
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 satisfies the following conditions: (i) each of A,A* is diagonalizable; (ii) there exists an ordering image of the eigenspaces of A such that A*Visubset of or equal toVi-1+Vi+Vi+1 for 0less-than-or-equals, slantiless-than-or-equals, slantd, where V-1=0 and Vd+1=0; (iii) there exists an ordering image of the eigenspaces of A* such that image for 0less-than-or-equals, slantiless-than-or-equals, slantδ, where image and image; (iv) there is no subspace W of V such that AWsubset of or equal toW, A*Wsubset of or equal toW, W≠0,W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and that for 0less-than-or-equals, slantiless-than-or-equals, slantd the dimensions of image coincide; we denote this common value by ρi. The sequence image is called the shape of the pair. In this paper we assume the shape is (1,2,1) and obtain the following results. We describe six bases for V; one diagonalizes A, another diagonalizes A*, and the other four underlie the split decompositions for A,A*. We give the action of A and A* on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape (1,2,1) in terms of a sequence of scalars called the parameter array.
