The partition algebra revisited
type: Item
instance of: scholarly article
title: The partition algebra revisited
zbMATH Open document ID: 0974.20013
author: William F. IV Doran
author: David Wales
published in: Journal of Algebra
publication date: 2001-02-28T00:00:00Z
review text: The partition algebra introduced by P.~Martin is a generalization of both the Temperley-Lieb algebra and the Brauer algebra. In the reviewer's paper [in Compos. Math. 119, No. 1, 99-109 (1999; Zbl 0939.16006)] it is proved that the partition algebra is also cellular in the sense of \textit{J.~J.~Graham} and \textit{G.~I.~Lehrer} [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)]. Given a natural number \(n\) and an element \(Q\) from some fixed field \(K\), the partition algebra \(P_n(Q)\) over the field \(K\) can be defined, and the cell modules are parametrized by integer partitions \(\lambda\) with \(|\lambda|\leq n\). When \(K\) is the field of complex numbers and \(Q\neq 0\), \textit{P.~Martin} [in J. Algebra 183, No. 2, 319-358 (1996; Zbl 0863.20009)] gives a condition for which there is a non-zero homomorphism between two cell modules \(\Delta_n(\lambda)\) and \(\Delta_n(\mu)\). In the present paper, the authors provide a more combinatorial view of Martin's result. Moreover, the authors not only extend Martin's results to include the case \(Q=0\), but also close a gap in the paper of Martin. This is a very powerful result.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16E60
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 1525274
zbMATH Keywords: partition algebras
zbMATH Keywords: symmetric groups
zbMATH Keywords: Temperley-Lieb algebras
zbMATH Keywords: Brauer algebras
zbMATH Keywords: cellular algebras
zbMATH Keywords: cell modules
reviewed by: Changchang Xi
MaRDI profile type: Publication
full work available at URL: jabr.2000.8365
OpenAlex ID: W1984483279
cites work: Q3995195
cites work: On the semisimplicity of the Brauer centralizer algebras
cites work: Cellular algebras
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cites work: The structure of the partition algebras
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cites work: Q4001520
cites work: Q3748279
DOI: 10.1006/JABR.2000.8365
description: scientific article; zbMATH DE number 1525274