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Idempotent systems
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In this paper we introduce the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. We focus on a family of idempotent systems, said to be symmetric. A symmetric idempotent system is an abstraction of the primary module for the subconstituent algebra of a symmetric association scheme. We describe the symmetric idempotent systems in detail. We also consider a class of symmetric idempotent systems, said to be
P
-polynomial and
Q
-polynomial. In the topic of orthogonal polynomials there is an object called a Leonard system. We show that a Leonard system is essentially the same thing as a symmetric idempotent system that is
P
-polynomial and
Q
-polynomial.
Title: Idempotent systems
Description:
In this paper we introduce the notion of an idempotent system.
This linear algebraic object is motivated by the structure of an association scheme.
We focus on a family of idempotent systems, said to be symmetric.
A symmetric idempotent system is an abstraction of the primary module for the subconstituent algebra of a symmetric association scheme.
We describe the symmetric idempotent systems in detail.
We also consider a class of symmetric idempotent systems, said to be
P
-polynomial and
Q
-polynomial.
In the topic of orthogonal polynomials there is an object called a Leonard system.
We show that a Leonard system is essentially the same thing as a symmetric idempotent system that is
P
-polynomial and
Q
-polynomial.
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