The structure of a tridiagonal pair
نویسندگان
چکیده
Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of K-linear transformations A : V → V and A : V → V that satisfy the following conditions: (i) each of A,A is diagonalizable; (ii) there exists an ordering {Vi} d i=0 of the eigenspaces of A such that A Vi ⊆ Vi−1+Vi+Vi+1 for 0 ≤ i ≤ d, where V−1 = 0 and Vd+1 = 0; (iii) there exists an ordering {V ∗ i } δ i=0 of the eigenspaces of A such that AV ∗ i ⊆ V ∗ i−1 +V ∗ i +V ∗ i+1 for 0 ≤ i ≤ δ, where V ∗ −1 = 0 and V ∗ δ+1 = 0; (iv) there is no subspace W of V such that AW ⊆ W , A W ⊆ W , W 6= 0, W 6= V . We call such a pair a tridiagonal pair on V . It is known that d = δ and for 0 ≤ i ≤ d the dimensions of Vi, Vd−i, V ∗ i , V ∗ d−i coincide. In this paper we show that the following (i)–(iv) hold provided that K is algebraically closed: (i) Each of V0, V ∗ 0 , Vd, V ∗ d has dimension 1. (ii) There exists a nondegenerate symmetric bilinear form 〈 , 〉 on V such that 〈Au, v〉 = 〈u,Av〉 and 〈Au, v〉 = 〈u,Av〉 for all u, v ∈ V . (iii) There exists a unique anti-automorphism of End(V ) that fixes each of A,A. (iv) The pair A,A is determined up to isomorphism by the data ({θi} d i=0; {θ ∗ i } d i=0; {ζi} d i=0), where θi (resp. θ ∗ i ) is the eigenvalue of A (resp. A ) on Vi (resp. V ∗ i ), and {ζi} d i=0 is the split sequence of A,A corresponding to {θi} d i=0 and {θ ∗ i } d i=0.
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