Brauer algebra

In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer[1] in the context of the representation theory of the orthogonal group.

It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality.

-algebra depending on the choice of a positive integer

is often specialised to the dimension of the fundamental representation of an orthogonal group

The Brauer algebra has the dimension A basis of

(that is, all perfect matchings of a complete graph

is obtained by concatenation: first identifying the endpoints in the bottom row of

(Figure AB in the diagram), 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 (Figure AB=nn in the diagram).

Thereby all closed loops in the middle of AB are removed.

of the basis elements is then defined to be the basis element corresponding to the new pairing multiplied by

satisfying the following relations: In this presentation

be the number of closed loops formed by identifying

is semisimple, they form a complete set of simple modules of

[4] These modules are parametrized by partitions, because they are built from the Specht modules of the symmetric group, which are themselves parametrized by partitions.

(This action can produce matchings that violate the restriction that

is A basis of this module is the set of elements

[5] The dimension is i.e. the product of a binomial coefficient, a double factorial, and the dimension of the corresponding Specht module, which is given by the hook length formula.

be a Euclidean vector space of dimension

acts by contraction followed by expansion in the

(The sum is in fact independent of the choice of this basis.)

This action is useful in a generalisation of the Schur-Weyl duality: if

It follows that the Brauer algebra has a natural action on the space of polynomials on

, which commutes with the action of the orthogonal group.

is a negative even integer, the Brauer algebra is related by Schur-Weyl duality to the symplectic group

The walled Brauer algebra

Diagrammatically, it consists of diagrams where the only allowed pairings are of the types

[6] The walled Brauer algebra is generated by

These generators obey the basic relations of

be the natural representation of the general linear group

, which is related by Schur-Weyl duality to the action of