Idempotent systems (Q2031620)

In this work, the notion of an idempotent system is introduced. The linear algebraic object is motivated by the structure of an association scheme. A symmetric idempotent system is an abstraction of the primary module for the subconstituent algebra of a symmetric association scheme. Symmetric idempotent systems are described in detail. In particular, a class of symmetric systems is considered to be \(P\)-polynomial and \(Q\)-polynomial. In the topic of orthogonal polynomials, there is an object called a Leonard system. It is shown that a Leonard system is essentially the same thing as a symmetric idempotent system that is \(P\)-polynomial and \(Q\)-polynomial.