The Brauer algebra is a -algebra depending on the choice of a positive integern. is an indeterminant, but in practice is often specialised to the dimension of the fundamental representation of an orthogonal group. The Brauer algebra has dimension and has a basis consisting ofall pairings on a set of elements . The elements are usually written in a row, with the elements beneath them. The product of two basis elements and is obtained by first identifying the endpoints in the bottom row of and the top row of , then deleting the endpoints in the middle row and joining endpoints in the remaining two rows if they are joined, directly or by a path, in AB . Thereby all closed loops in the middle of AB are removed. The product of the basis elements is then defined to be the basis element corresponding to the new pairing multiplied by where is the number of deleted loops. In the example.
can also be defined as the -algebra with generators satisfying the following relations:
Relations of the symmetric group:
Almost-idempotent relation:
Commutation:
Tangle relations
Untwisting:
In this presentation represents the diagram in which is always connected to directly beneath it except for and which are connected to ans respectively. Similarly represents the diagram in which is always connected to directly beneath it except for being connected to and to.
Properties
The subalgebra generated by the is the group algebra of the symmetric group. The Brauer algebra is a cellular algebra.
Let be a euclidean vector space of dimension. Then write for the specialisation where acts on by multiplication with. The tensor power is naturally a -module: acts by switching the th and th tensor factor and acts by contraction followed by expansion in the th and th tensor factor, i.e. acts as where is any orthonormal basis of . This action is useful in a generalisation of the Schur-Weyl duality: The image of inside is exactly the centraliser of inside and vice versa. The tensor power is therefore both an - and a -module and satisfies where runs over certain partitions and are the irreducible - and -module associated with respectively.
The orthogonal group
If Od is the orthogonal group acting on V = Rd, then the Brauer algebra has a natural action on the space of polynomials on Vn commuting with the action of the orthogonal group.
