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Matrix units associated with the split basis of a Leonard pair

Research paper by Kazumasa Nomura, Paul Terwilliger

Indexed on: 19 Feb '06Published on: 19 Feb '06Published in: Mathematics - Rings and Algebras



Abstract

Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy (i), (ii) below: (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. We call such a pair a {\em Leonard pair} on $V$. It is known that there exists a basis for $V$ with respect to which the matrix representing $A$ is lower bidiagonal and the matrix representing $A^*$ is upper bidiagonal. In this paper we give some formulae involving the matrix units associated with this basis.