Towards a classification of the tridiagonal pairs Original Research Article

Let image denote a field and let V denote a vector space over image with finite positive dimension. Let End(V) denote the image-algebra consisting of all image-linear transformations from V to V. We consider a pair A,A*set membership, variantEnd(V) that satisfy (i)–(iv) below: (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. Let image denote the element of End(V) such that image and image for 1less-than-or-equals, slantiless-than-or-equals, slantd. Let image (resp. image) denote the image-subalgebra of End(V) generated by A (resp. A*). In this paper we prove that the span of image equals the span of image, and that the elements of image mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs.