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

TL;DR

This paper introduces the concept of tridiagonal pairs, a generalization of Leonard pairs, which are pairs of linear transformations with specific diagonal and tridiagonal matrix representations, relevant to $P$- and $Q$-polynomial association schemes.

Contribution

The paper defines tridiagonal pairs as a broader class than Leonard pairs, expanding the framework for studying algebraic structures related to association schemes.

Findings

Introduced the concept of tridiagonal pairs.

Established the relationship between Leonard pairs and tridiagonal pairs.

Provided foundational properties of tridiagonal pairs.

Abstract

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.