  The partition algebra is an associative algebra with a basis of set-partition
diagrams and multiplication given by diagram concatenation. It contains as
subalgebras a large class of diagram algebras including the Brauer, planar
partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook
monoid, and symmetric group algebras. We give a construction of the irreducible
modules of these algebras in two isomorphic ways: first, as the span of
symmetric diagrams on which the algebra acts by conjugation twisted with an
irreducible symmetric group representation and, second, on a basis indexed by
set-partition tableaux such that diagrams in the algebra act combinatorially on
tableaux. The first representation is analogous to the Gelfand model and the
second is a generalization of Young's natural representation of the symmetric
group on standard tableaux. The methods of this paper work uniformly for the
partition algebra and its diagram subalgebras. As an application, we express
the characters of each of these algebras as nonnegative integer combinations of
symmetric group characters whose coefficients count fixed points under
conjugation.
