Balanced Leonard pairs

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 (respectively v0, v1, … , vd) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ⩽ i ⩽ d, let ai denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of vi, when we write A∗vi as a linear combination of v0, v1, … , vd.In this paper we show a0 = ad if and only if . Moreover we show that for d ⩾ 1 the following are equivalent; (i) a0 = ad and a1 = ad−1; (ii) and ; (iii) ai = ad−i and for 0 ⩽ i ⩽ d. These give a proof of a conjecture by the second author. We say A, A∗ is balanced whenever ai = ad−i and for 0 ⩽ i ⩽ d. We say A,A∗ is essentially bipartite (respectively essentially dual bipartite) whenever ai (respectively ) 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.