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Pseudo-direct sums and wreath products of loose-coherent algebras with applications to coherent configurations. (English) Zbl 1368.16038
Summary: We introduce the notion of a loose-coherent algebra, which is a special semisimple subalgebra of the matrix algebra, and define two operations to obtain new loose-coherent algebras from the old ones: the pseudo-direct sum and the wreath product. For two arbitrary coherent configurations \(\mathfrak{C}\), \(\mathfrak{D}\) and their wreath product \(\mathfrak{C} \wr \mathfrak{D}\), it is difficult to express the Terwilliger algebra \(\mathcal{T}_{(x, y)}(\mathfrak{C} \wr \mathfrak{D})\) in terms of the Terwilliger algebras \(\mathcal{T}_x(\mathfrak{C})\) and \(\mathcal{T}_y(\mathfrak{D})\). By using the concept and operations of loose-coherent algebras, we find a very simple such expression. As a direct consequence of this expression, we obtain the central primitive idempotents of \(\mathcal{T}_{(x, y)}(\mathfrak{C} \wr \mathfrak{D})\) in terms of the central primitive idempotents of \(\mathcal{T}_x(\mathfrak{C})\) and \(\mathcal{T}_y(\mathfrak{D})\). Many results in are special cases of the results in this paper.

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