Modular Leonard triples
Let \(\mathbb{K}\) denote a field. A square matrix over \(\mathbb{K}\) is said to be tridiagonal whenever every nonzero entry appears on the diagonal, the superdiagonal, or the subdiagonal. A tridiagonal matrix is irreducible whenever the entries on the sub- and superdiagonals are all nonzero. Let \(V\) denote a vector space over \(\mathbb{K}\) of finite positive dimension. An ordered triple \(A\), \(A^*\), \(A^\diamondsuit\) of linear operators on \(V\) is said to be a Leonard triple whenever for each \(B\in\{A, A^*, A^\diamondsuit\}\), there exists a basis of \(V\) with respect to which the matrix representing \(B\) is diagonal and the matrices representing the other two operators are irreducible tridiagonal. A Leonard triple \(A\), \(A^*\), \(A^\diamondsuit\) is said to be modular whenever for each \(B\in\{A,A^*,A^\diamondsuit\}\), there exists an antiautomorphism of \(\text{End}(V)\) which fixes \(B\) and swaps the other two operators. The paper classifies the modular Leonard triples up to isomorphism.
scientific article; zbMATH DE number 5167493
scientific article