Balanced Leonard pairs
Let \(V\) denote a finite-dimensional vector space over any field, dim\(\,V= d+1>0\). A Leonard pair \((A, A^*)\) is a pair of linear transformations \(A: V\to V\) and \(A^*: V\to V\) with the following properties: (i) There exists a basis \((v_i^*)\) of \(V\) such that the matrices of \(A\) and \(A^*\) with respect to this basis are irreducible tridiagonal and diagonal, respectively. (ii) There exists a basis \((v_i)\) of \(V\) such that the matrices of \(A^*\) and \(A\) with respect to this basis are irreducible tridiagonal and diagonal, respectively. Denote by \(a_i\) the \(i\)th entry in the main diagonal of the tridiagonal matrix representing \(A\) with respect to \((v_i^*)\), where \(0\leq i\leq d\). The scalars \(a_i^*\) are defined accordingly. The Leonard pair \((A, A^*)\) is said to be balanced if \(a_i=a_{d-i}\) and \(a_i^*=a_{d-i}^*\) for \(0\leq i\leq d\). The Leonard pair \((A, A^*)\) is said to be essentially bipartite (essentially dual bipartite) if \(a_i\) (\(a_i^*\)) is independent of \(i\) for \(0\leq i\leq d\). The authors show that \(a_0=a_d\) if, and only if, \(a_0^*=a_d^*\). Furthermore, the following assertions are shown to be equivalent: (i) \(a_0=a_d\) and \(a_1=a_{d-1}\); (ii) \(a_0^*=a_d^*\) and \(a_1^*=a_{d-1}^*\); and (iii) \((A, A^*)\) is balanced. For \(d\not=2\), the authors also prove that if \((A,A^*)\) is balanced then the pair \((A, A^*)\) is essentially bipartite or dual bipartite.
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