Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
Terwilliger, Paul
arXiv, 0306291 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2003-06-19
Classification:  Mathematics - Rings and Algebras,  Mathematical Physics,  Mathematics - Combinatorics,  Mathematics - Quantum Algebra,  Mathematics - Representation Theory,  17B37,  05E30, 33C45, 33D45 
@article{0306291,
     author = {Terwilliger, Paul},
     title = {Two linear transformations each tridiagonal with respect to an
  eigenbasis of the other; comments on the parameter array},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0306291}
}

Terwilliger, Paul. Two linear transformations each tridiagonal with respect to an
  eigenbasis of the other; comments on the parameter array. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0306291/