Mathematics – Rings and Algebras
Scientific paper
2003-06-19
Mathematics
Rings and Algebras
26 pages
Scientific paper
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (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 diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. The structure of any given Leonard pair is deterined by a certain sequence of scalars called its {\it parameter array}. The set of parameter arrays is an affine algebraic variety. We give two characterizations of this variety. One involves bidiagonal matrices and the other involves orthogonal polynomials.
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