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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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##### References:
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