Some algebra related to $P$-and $Q$-polynomial association schemes

Details Some algebra related to $P$-and $Q$-polynomial association schemes Some algebra related to $P$-and $Q$-polynomial association schemes

2004-06-27

arxiv.org/abs/math/0406556v1

Proceedings of DIMACS conference on Codes and Association Schemes, (Piscataway NJ, 1999), 167--192. Amer. Math. Soc. Providenc

Mathematics

Combinatorics

26 pages

Scientific paper

Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. Consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal. (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. Such a pair is called a Leonard pair on $V$. In this paper we introduce a mild generalization of a Leonard pair called a tridiagonal pair. A Leonard pair is the same thing as a tridiagonal pair such that for each transformation all eigenspaces have dimension one.

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