# ALGEBRAIC COMBINATORICS

Idempotent systems
Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 329-357.

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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.159
Classification: 17B37,  15A21
Keywords: idempotent system, association scheme, Leonard pair
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Nomura, Kazumasa; Terwilliger, Paul. Idempotent systems. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 329-357. doi : 10.5802/alco.159. https://alco.centre-mersenne.org/articles/10.5802/alco.159/

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