\(G\)-colored partition algebras as centralizer algebras of wreath products
The partition algebra \(P_k(x)\) was introduced independently by \textit{V. F. R. Jones} [in Subfactors 1993, 259-267 (1994; Zbl 0938.20505)] and by \textit{P. Martin} [J. Knot Theory Ramifications 3, No. 1, 51-82 (1994; Zbl 0804.16002)]. In particular, Jones proved that \(P_k(n)\) is the algebra of all transformations commuting with the action of the symmetric group \(S_n\) on the tensor product \(V^{\otimes^k}\) of its standard permutation representation on \(V=\mathbb{C}^n\). In the present paper, for every group \(G\), the author introduces a \(G\)-colored partition algebra \(P_k(x;G)\); it is defined using diagrams with oriented edges labeled by elements of \(G\). For \(G\) finite, he shows that \(P_k(x;G)\) is the centralizer algebra of an action of the wreath product \(G\wr S_n\) on tensor products of its permutation representation on \(\{1,2,\dots,n\}\times G\).
scientific article; zbMATH DE number 1963616
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