The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation.
Representations of the partition algebra are built from sets of diagrams and from representations of the symmetric group.
is represented as a diagram, with lines connecting elements in the same subset.
-basis made of partitions, and a multiplication given by diagram concatenation.
is the number of connected components that are disconnected from the top and bottom elements.
These generators obey relations that include[2] Other elements that are useful for generating subalgebras include
In terms of the original generators, these elements are The partition algebra
[1] The partition algebra is finite-dimensional, with
Subalgebras of the partition algebra can be defined by the following properties:[3] Combining these properties gives rise to 8 nontrivial subalgebras, in addition to the partition algebra itself:[1][3] The symmetric group algebra
Inclusions of planar into non-planar algebras: Inclusions from constraints on subset size: Inclusions from allowing top-top and bottom-bottom lines: We have the isomorphism: In addition to the eight subalgebras described above, other subalgebras have been defined: An algebra with a half-integer index
[2] Periodic subalgebras are generated by diagrams that can be drawn on an annulus without line crossings.
Such subalgebras include a translation element
The translation element and its powers are the only combinations of
Such partitions are represented by diagrams with no top-top lines, with at least one line for each top element.
can be described combinatorially in terms of set-partition tableaux: Young tableaux whose boxes are filled with the blocks of a set partition.
is irreducible, and the set of irreducible finite-dimensional representations of the partition algebra is Representations of non-planar subalgebras have similar structures as representations of the partition algebra.
For example, the Brauer-Specht modules of the Brauer algebra are built from Specht modules, and certain sets of partitions.
In the case of the planar subalgebras, planarity prevents nontrivial permutations, and Specht modules do not appear.
For example, a standard module of the Temperley–Lieb algebra is parametrized by an integer
, and a basis is simply given by a set of partitions.
The following table lists the irreducible representations of the partition algebra and eight subalgebras.
are indexed by sequences of partitions.
, there is a natural action of the partition algebra
This action is defined by the matrix elements of a partition
:[2] This matrix element is one if all indices corresponding to any given partition subset coincide, and zero otherwise.
to be the natural permutation representation of the symmetric group
, the tensor product space decomposes into irreducible representations as[1] where
The duality between the symmetric group and the partition algebra generalizes the original Schur-Weyl duality between the general linear group and the symmetric group.
In the relevant tensor product spaces, we write
-dimensional representation of the first group or algebra: