Partition Algebras

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)

Set-partition tableaux and representations of diagram algebras

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