  The partition algebras are algebras of diagrams (which contain the group
algebra of the symmetric group and the Brauer algebra) such that the
multiplication is given by a combinatorial rule and such that the structure
constants of the algebra depend polynomially on a parameter. This is a survey
paper which proves the primary results in the theory of partition algebras.
Some of the results in this paper are new. This paper gives: (a) a presentation
of the partition algebras by generators and relations, (b) shows that each
partition algebra has an ideal which is isomorphic to a basic construction and
such that the quotient is isomorphic to the group algebra of the symmetric
gropup, (c) shows that partition algebras are in "Schur-Weyl duality" with the
symmetric groups on tensor space, (d) provides a construction of "Specht
modules" for the partition algebras (integral lattices in the generic
irreducible modules), (e) determines (with a couple of exceptions) the values
of the parameter where the partition algebras are semisimple, (f) provides
"Murphy elements" for the partition algebras that play exactly analogous roles
to the classical Murphy elements for the group algebra of the symmetric group.
The primary new results in this paper are (a) and (f).
