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Terwilliger algebras of wreath products of association schemes. (English) Zbl 1329.05313
Summary: The Terwilliger algebra of an association scheme of order \(n\) introduced in [P. Terwilliger, J. Algebr. Comb. 1, No. 4, 363–388 (1992; Zbl 0785.05089)] is a subalgebra of the matrix algebra of all \(n \times n\) matrices. Terwilliger algebras of wreath products of special association schemes are studied in several papers. In this paper we study the Terwilliger algebra of the wreath product \(T \wr S\) of two arbitrary association schemes \(S\) and \(T\). We will express the Terwilliger algebra of \(T \wr S\) and its primitive central idempotents in terms of the Terwilliger algebras of \(S\) and \(T\) and their primitive central idempotents. The known results of A. Hanaki et al. [J. Algebra 343, No. 1, 195–200 (2011; Zbl 1235.05157)], K. Kim [Linear Algebra Appl. 437, No. 11, 2773–2780 (2012; Zbl 1253.05147)], etc. are special cases of our results.

MSC:
05E30 Association schemes, strongly regular graphs
16S50 Endomorphism rings; matrix rings
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