Parvathi, M.; Kennedy, A. Joseph Representations of vertex colored partition algebras. (English) Zbl 1081.20010 Southeast Asian Bull. Math. 28, No. 3, 493-518 (2004). The authors use the representation theory of the partition algebra [cf. M. Bloss, Dissertation, Univ. Wisconsin, Madison (2002) and V. F. R. Jones, Proceedings of the Taniguchi symposium on operator algebras, 259-267 (1994; Zbl 0938.20505)] to index the inequivalent irreducible representations of the \(G\)-vertex colored partition algebras \(P_k(x,G)\) [cf. M. Parvathi, A. J. Kennedy, Commun. Algebra 32, No. 11, 4337-4361 (2004; see the preceeding review Zbl 1081.20009)] and the extended \(G\)-vertex colored partition algebras \(\widehat P_k(x,G)\) [cf. M. Parvathi, A. J. Kennedy, Algebra Discrete Math. 2005, No. 2, 58-79 (2005; Zbl 1091.20005)], and to compute their dimensions, in the semisimple cases. For the towers \(P_{k-1}(x,G)\subseteq P_k(x,G)\) and \(\widehat P_{k-1}(x,G)\subseteq\widehat P_k(x,G)\), they describe the Bratteli diagrams and the branching rules. Reviewer: Yichuan Yang (Stuttgart) Cited in 2 Documents MSC: 20C15 Ordinary representations and characters 05E10 Combinatorial aspects of representation theory 20C30 Representations of finite symmetric groups 16G20 Representations of quivers and partially ordered sets 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 20G05 Representation theory for linear algebraic groups Keywords:partition algebras; centralizer algebras; direct products; wreath products; symmetric groups; permutation representations; tensor products; permutation modules; irreducible representations; branching rules Citations:Zbl 0938.20505; Zbl 1081.20009; Zbl 1091.20005 PDFBibTeX XMLCite \textit{M. Parvathi} and \textit{A. J. Kennedy}, Southeast Asian Bull. Math. 28, No. 3, 493--518 (2004; Zbl 1081.20010)