ALGEBRAIC COMBINATORICS

no. 2
Idempotent systems
Nomura, Kazumasa1; Terwilliger, Paul2
1 Tokyo Medical and Dental University Kohnodai, Ichikawa 272-0827, Japan
2 University of Wisconsin Dept. of mathematics 480 Lincoln Drive Madison, WI 53706 USA
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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.159
Classification: 17B37, 15A21
Keywords: idempotent system, association scheme, Leonard pair
Nomura, Kazumasa 1; Terwilliger, Paul 2

1 Tokyo Medical and Dental University Kohnodai, Ichikawa 272-0827, Japan
2 University of Wisconsin Dept. of mathematics 480 Lincoln Drive Madison, WI 53706 USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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