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 -polynomial and -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 -polynomial and -polynomial.
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Keywords: idempotent system, association scheme, Leonard pair
@article{ALCO_2021__4_2_329_0, author = {Nomura, Kazumasa and Terwilliger, Paul}, title = {Idempotent systems}, journal = {Algebraic Combinatorics}, pages = {329--357}, publisher = {MathOA foundation}, volume = {4}, number = {2}, year = {2021}, doi = {10.5802/alco.159}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.159/} }
Nomura, Kazumasa; Terwilliger, Paul. Idempotent systems. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 329-357. doi : 10.5802/alco.159. http://archive.numdam.org/articles/10.5802/alco.159/
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