Sharp tridiagonal pairs 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 image-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 for 0less-than-or-equals, slantiless-than-or-equals, slantd the dimensions of image coincide. We say the pair A,A* is sharp whenever dimV0=1. A conjecture of Tatsuro Ito and the second author states that if image is algebraically closed then A,A* is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A,A* is sharp and using the data image we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by image or imageor image. We prove that if the isomorphism class of Φ is determined by the parameter array then there exists a nondegenerate symmetric bilinear form left angle bracket,right-pointing angle bracket on V such that left angle bracketAu,vright-pointing angle bracket=left angle bracketu,Avright-pointing angle bracket and left angle bracketA*u,vright-pointing angle bracket=left angle bracketu,A*vright-pointing angle bracket for all u,vset membership, variantV.