Partition algebra
The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. Its subalgebras include diagram algebras such as the Brauer algebra, the Temperley–Lieb algebra, or the group algebra of the symmetric group. Representations of the partition algebra are built from sets of diagrams and from representations of the symmetric group. Definition

Diagrams 

A partition of  elements labelled  is represented as a diagram, with lines connecting elements in the same subset. In the following example, the subset  gives rise to the lines , and could equivalently be represented by the lines  (for instance). For  and , the partition algebra  is defined by a -basis made of partitions, and a multiplication given by diagram concatenation. The concatenated diagram comes with a factor , where  is the number of connected components that are disconnected from the top and bottom elements. Generators and relations 

The partition algebra  is generated by  elements of the type

These generators obey relations that include

Other elements that are useful for generating subalgebras include

In terms of the original generators, these elements are

Properties 

The partition algebra  is an associative algebra. It has a multiplicative identity

The partition algebra  is semisimple for . For any two  in this set, the algebras  and  are isomorphic. The partition algebra is finite-dimensional, with  (a Bell number). Subalgebras

Eight subalgebras 

Subalgebras of the partition algebra can be defined by the following properties:
 Whether they are planar i.e. whether lines may cross in diagrams. Whether subsets are allowed to have any size , or size , or only size . Whether we allow top-top and bottom-bottom lines, or only top-bottom lines. In the latter case, the parameter  is absent, or can be eliminated by . Combining these properties gives rise to 8 nontrivial subalgebras, in addition to the partition algebra itself:

The symmetric group algebra  is the group ring of the symmetric group  over . The Motzkin algebra is sometimes called the dilute Temperley–Lieb algebra in the physics literature. Properties 

The listed subalgebras are semisimple for . 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:

More subalgebras 

In addition to the eight subalgebras described above, other subalgebras have been defined:
 The totally propagating partition subalgebra  is generated by diagrams whose blocks all propagate, i.e. partitions whose subsets all contain top and bottom elements. These diagrams from the dual symmetric inverse monoid, which is generated by . The quasi-partition algebra  is generated by subsets of size at least two. Its generators are  and its dimension is . The uniform block permutation algebra  is generated by subsets with as many top elements as bottom elements. It is generated by . An algebra with a half-integer index  is defined from partitions of  elements by requiring that  and  are in the same subset. For example,  is generated by  so that , and . Periodic subalgebras are generated by diagrams that can be drawn on an annulus without line crossings. Such subalgebras include a translation element  such that . The translation element and its powers are the only combinations of  that belong to periodic subalgebras. Representations

Structure 

For an integer , let  be the set of partitions of  elements  (bottom) and  (top), such that no two top elements are in the same subset, and no top element is alone. Such partitions are represented by diagrams with no top-top lines, with at least one line for each top element. For example, in the case :

Partition diagrams act on  from the bottom, while the symmetric group  acts from the top. For any Specht module  of  (with therefore ), we define the representation of 

The dimension of this representation is

where  is a Stirling number of the second kind,  is a binomial coefficient, and  is given by the hook length formula. A basis of  can be described combinatorially in terms of set-partition tableaux: Young tableaux whose boxes are filled with the blocks of a set partition. Assuming that  is semisimple, the representation  is irreducible, and the 
set of irreducible finite-dimensional representations of the partition algebra is

Representations of subalgebras 

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  with , 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. The irreducible representations of  are indexed by partitions such that  and their dimensions are . The irreducible representations of  are indexed by partitions such that . The irreducible representations of  are indexed by sequences of partitions. Schur-Weyl duality 

Assume . For  a -dimensional vector space with basis , there is a natural action of the partition algebra  on the vector space . This action is defined by the matrix elements of a partition  in the basis :

This matrix element is one if all indices corresponding to any given partition subset coincide, and zero otherwise. For example, the action of a Temperley–Lieb generator is

Duality between the partition algebra and the symmetric group 

Let  be integer. Let us take  to be the natural permutation representation of the symmetric group . This -dimensional representation is a sum of two irreducible representations: the standard and trivial representations, . Then the partition algebra  is the centralizer of the action of  on the tensor product space , 

Moreover, as a bimodule over , the tensor product space decomposes into irreducible representations as 

where  is a Young diagram of size  built by adding a first row to , and  is the corresponding Specht module of . Dualities involving subalgebras 

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. There are other generalizations. In the relevant tensor product spaces, we write  for an irreducible -dimensional representation of the first group or algebra:

References

Further reading

Representation theory
Diagram algebras