In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer 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. Structure The Brauer algebra is a -algebra depending on the choice of a positive integer . Here is an indeterminate, but in practice is often specialised to the dimension of the fundamental representation of an orthogonal group . The Brauer algebra has the dimension Diagrammatic definition A basis of consists of all pairings on a set of elements (that is, all perfect matchings of a complete graph : any two of the elements may be matched to each other, regardless of their symbols). The elements are usually written in a row, with the elements beneath them. The product of two basis elements and is obtained by concatenation: first identifying the endpoints in the bottom row of and the top 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. 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 . Generators and relations can also be defined as the -algebra with generators satisfying the following relations: Relations of the symmetric group: whenever Almost-idempotent relation: Commutation: whenever 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 and respectively. Similarly represents the diagram in which is always connected to directly beneath it except for being connected to and to . Basic properties The Brauer algebra is a subalgebra of the partition algebra. The Brauer algebra is semisimple if . The subalgebra of generated by the generators is the group algebra of the symmetric group . The subalgebra of generated by the generators is the Temperley-Lieb algebra . The Brauer algebra is a cellular algebra. For a pairing let be the number of closed loops formed by identifying with for any : then the Jones trace obeys i.e. it