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Combinatorial Gelfand models for semisimple diagram algebras. (English) Zbl 1314.16009

From the introduction: We construct combinatorial (involutory) Gelfand models for the following diagram algebras in the case when they are semi-simple: Brauer algebras, their partial analogues, walled Brauer algebras, their partial analogues, Temperley-Lieb algebras, their partial analogues, walled Temperley-Lieb algebras, their partial analogues, partition algebras and their Temperley-Lieb analogues.
The main result of the paper, which gives an explicit combinatorial Gelfand model for all diagram algebras mentioned above in the semi-simple case, is formulated and proved in Section 3 after preliminaries on diagram algebras and their representations that are collected in Section 2. For the Brauer algebra this Gelfand model was considered by M. Schramm [in Combinatorial Gelfand models. Master Thesis. Uppsala University (2012)].
Shortly after publication of the preprint version of this paper it turned out that the same results were independently obtained by T. Halverson and M. Reeks [in J. Algebr. Comb. 41, No. 2, 229-255 (2015; Zbl 1308.05107)].

MSC:

16G20 Representations of quivers and partially ordered sets
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20C08 Hecke algebras and their representations
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras

Citations:

Zbl 1308.05107
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References:

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