\(G\)-colored partition algebras as centralizer algebras of wreath products
type: Item
instance of: scholarly article
title: \(G\)-colored partition algebras as centralizer algebras of wreath products
zbMATH Open document ID: 1028.20007
DOI: 10.1016/S0021-8693(03)00132-7
author: Matthew Bloss
published in: Journal of Algebra
publication date: 2003-08-13T00:00:00Z
review text: 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 \(\{1,2,\dots,n\}\times G\).
reviewed by: Fabio Scarabotti
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 1963616
zbMATH Keywords: partition algebras
zbMATH Keywords: centralizer algebras
zbMATH Keywords: wreath products
zbMATH Keywords: symmetric groups
zbMATH Keywords: permutation representations
zbMATH Keywords: tensor products
MaRDI profile type: Publication
cites work: Q4392287
cites work: Characters of the partition algebras.
cites work: Q4245140
cites work: The algebra of \(G\)-relations
cites work: TEMPERLEY-LIEB ALGEBRAS FOR NON-PLANAR STATISTICAL MECHANICS — THE PARTITION ALGEBRA CONSTRUCTION
cites work: On an algebraic approach to higher dimensional statistical mechanics
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full work available at URL: s0021-8693(03)00132-7
OpenAlex ID: W2049647042
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description: scientific article; zbMATH DE number 1963616