Balanced Leonard pairs

Let denote a field, and let V denote a vector space over 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.