  A Gelfand model for a semisimple algebra A over C is a complex linear
representation that contains each irreducible representation of 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 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.
