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Representations of vertex colored partition algebras. (English) Zbl 1081.20010

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.

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
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