Balanced Leonard Pairs

الأصل · EN

Let K denote a field and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A:V → V and A*:V → V that satisfy the following two conditions: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. Let v*₀,..., v*d (resp. v₀,..., vd) denote a basis for V that satisfies (i) (resp. (ii)). For 0 ≤ i ≤ d let aᵢ denote the coefficient of v*ᵢ, when we write A v*ᵢ as a linear combination of v*₀,..., v*d, and let a*ᵢ denote the coefficient of vᵢ, when we write A* vᵢ as a linear combination of v₀..., vd. In this paper we show a₀=ad if and only if a*₀=a*d. Moreover we show that for d ≥ 1 the following are equivalent: (i) a₀=ad and a₁=ad₋₁; (ii) a*₀=a*d and a*₁=a*d₋₁; (iii) aᵢ=ad₋ᵢ and a*ᵢ=a*d₋ᵢ for 0 ≤ i ≤ d. We say A, A* is balanced whenever (i)--(iii) hold. We say A, A* is essentially bipartite (resp. essentially dual bipartite) whenever aᵢ (resp. a*ᵢ) is independent of i for 0 ≤ i ≤ d. Observe that if A, A* is essentially bipartite or dual bipartite, then A, A* is balanced. For d ≠ 2 we show that if A, A* is balanced then A, A* is essentially bipartite or dual bipartite.