Gelfand Models for Diagram Algebras

Abstract : A Gelfand model for a semisimple algebra $\mathsf{A}$ over $\mathbb{C}$ is a complex linear representation that contains each irreducible representation of $\mathsf{A}$ with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of combinatorial diagram algebras including: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid algebras. In each case, the model representation is given by diagrams acting via ``signed conjugation" on the linear span of their vertically symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group, and, in fact, our method is to use the Jones basic construction to lift the Saxl model from the symmetric group to each diagram algebra. In the case of the planar diagram algebras, our construction exactly produces the irreducible representations of the algebra.
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Alain Goupil and Gilles Schaeffer. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp.837-848, 2013, DMTCS Proceedings
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Tom Halverson. Gelfand Models for Diagram Algebras. Alain Goupil and Gilles Schaeffer. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp.837-848, 2013, DMTCS Proceedings. 〈hal-01229664〉

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