Geodesic
Gelfand models for diagram algebras
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 2, pp. 229-255.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

A Gelfand model for a semisimple algebra $\mathsf A$ over an algebraically closed field $\mathbb K$ is a 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 semisimple, 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 horizontally symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group. 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.
Classification : 05E10, 16G99, 20C30
Keywords: Gelfand model, multiplicity-free representation, symmetric group, partition algebra, Brauer algebra, Temperley-Lieb algebra, Motzkin algebra, rook monoid 
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Halverson, Tom; Reeks, Mike. Gelfand models for diagram algebras. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 2, pp. 229-255. http://geodesic.mathdoc.fr/item/JAC_2015__41_2_a11/