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A Gelfand model for a semisimple algebra A over C is a complex linearrepresentation that contains each irreducible representation of A withmultiplicity exactly one. We give a method of constructing these models thatworks uniformly for a large class of semisimple, combinatorial diagram algebrasincluding: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb,Motzkin, and planar rook monoid algebras. In each case, the modelrepresentation is given by diagrams acting via "signed conjugation" on thelinear span of their vertically symmetric diagrams. This representation is ageneralization of the Saxl model for the symmetric group, and, in fact, ourmethod is to use the Jones basic construction to lift the Saxl model from thesymmetric group to each diagram algebra. In the case of the planar diagramalgebras, our construction exactly produces the irreducible representations ofthe algebra.

Gelfand Models for Diagram Algebras