How to Sharpen a Tridiagonal Pair

Abstract

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of 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 orderingWe call such a pair a tridiagonal pair on V . It is known that d = δ, and forDenote this common dimension by ρ i and call A, A * sharp whenever ρ 0 = 1. Let T denote the F-subalgebra of End F (V ) generated by A, A * . We show: (i) the center Z(T ) is a field whose dimension over F is ρ 0 ; (ii) the field Z(T ) is isomorphic to each of, where E i (resp. E * i ) is the primitive idempotent of A (resp. A * ) associated with V i (resp. V * i ); (iii) with respect to the Z(T )-vector space V the pair A, A * is a sharp tridiagonal pair.Keywords. Tridiagonal pair, Tridiagonal system. 2000 Mathematics Subject Classification. Primary: 15A21. Secondary: 05E30, 05E35. Tridiagonal pairsThroughout this paper F denotes a field and V denotes a vector space over F with finite positive dimension.We begin by recalling the notion of a tridiagonal pair. We will use the following terms. Let End(V ) = End F (V ) denote the F-algebra of all F-linear transformations from V to V . Given A ∈ End(V ) and a subspace W ⊆ V , we call W an eigenspace of A whenever W = 0 and there exists θ ∈ F such that W = {v ∈ V |Av = θv}; in this case θ is the eigenvalue of A associated with W . We say that A is diagonalizable whenever V is spanned by the eigenspaces of A.

Totally bipartite tridiagonal pairs

“…By [71,Theorem 4.6], the elements A, A * act on {U i } d i=0 as follows: [74,97,98,106,108,112,113,140,142,157]. Some miscellaneous results about tridiagonal pairs and systems can be found in [1,4,28,85,87].…”

A classification of sharp tridiagonal pairs

“…By [59,Theorem 1.3] we have ρ i ≤ ρ 0 d i for 0 ≤ i ≤ d. We call the sequence {ρ i } d i=0 the shape of A, A * . See [28,38,46,47,55,59] for results on the shape. The TD pair A, A * is called sharp whenever ρ 0 = 1.…”

“…By [57,Theorem 1.3], if F is algebraically closed then A, A * is sharp. In any case A, A * can be "sharpened" by replacing F with a certain field extension K of F that has index [K : F] = ρ 0 [38,Theorem 4.12]. Suppose that A, A * is sharp.…”

“…Define an F-algebra homomorphism σ : P → P that sends x 0 → 1 − d i=0 x i and fixes x j for 1 ≤ j ≤ d. The composition of σ with itself is the identity, so σ is an isomorphism. To obtain (37), apply σ to each side of (38) and note that σ sends K 0 to K while leaving F[x 1 , . .…”

Two commuting operators associated with a tridiagonal pair

“…The concept of a TD pair originated in the theory of Q-polynomial distance-regular graphs [17]. Since that beginning the TD pairs have been investigated in a systematic way; for notable papers along this line see [1,2,3,4,5,6,7,8,9,10,15,18]. Several of these papers focus on a class of TD pair said to be sharp.…”