Brauer algebra

In mathematics, a Brauer algebra is an algebra introduced by used in 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.
Definition.
In terms of diagrams.
The Brauer algebra formula_1 is a formula_2-algebra depending on the choice of a positive integer "n". Here formula_3 is an indeterminant, but in practice formula_3 is often specialised to the dimension of the fundamental representation of an orthogonal group formula_5. The Brauer algebra has dimension formula_6 and has a basis consisting of all pairings on a set of formula_7 elements formula_8 (that is, all perfect matchings of a complete graph formula_9: any two of the formula_7 elements may be matched to each other, regardless of their symbols). The elements formula_11 are usually written in a row, with the elements formula_12 beneath them. The product of two basis elements formula_13 and formula_14 is obtained by first identifying the endpoints in the bottom row of formula_13 and the top row of formula_14 (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 formula_17 of the basis elements is then defined to be the basis element corresponding to the new pairing multiplied by formula_18 where formula_19 is the number of deleted loops. In the example formula_20.
In terms of generators and relations.
formula_1 can also be defined as the formula_2-algebra with generators formula_23 satisfying the following relations:
In this presentation formula_37 represents the diagram in which formula_38 is always connected to formula_39 directly beneath it except for formula_11 and formula_41 which are connected to formula_42 ans formula_12 respectively. Similarly formula_44 represents the diagram in which formula_38 is always connected to formula_39 directly beneath it except for formula_11 being connected to formula_41 and formula_12 to formula_42.
Properties.
The subalgebra generated by the formula_37 is the group algebra of the symmetric group. The Brauer algebra is a cellular algebra.
Action on tensor powers.
Let formula_52 be a euclidean vector space of dimension formula_53. Then write formula_54 for the specialisation formula_55 where formula_3 acts on formula_57 by multiplication with formula_53. The tensor power formula_59 is naturally a formula_54-module: formula_37 acts by switching the formula_62th and formula_63th tensor factor and formula_44 acts by contraction followed by expansion in the formula_62th and formula_63th tensor factor, i.e. formula_44 acts as
where formula_69 is any orthonormal basis of formula_52 (the sum is in fact independent of the choice of such a basis).
This action is useful in a generalisation of the Schur-Weyl duality: The image of formula_54 inside formula_72 is exactly the centraliser of formula_73 inside formula_72 and vice versa. The tensor power formula_75 is therefore both an formula_73- and a formula_54-module and satisfies
where formula_79 runs over certain partitions and formula_80 are the irreducible formula_73- and formula_54-module associated with formula_79 respectively.
The orthogonal group.
If "O""d"(R) is the orthogonal group acting on "V" = R"d", then 
the Brauer algebra has a natural action on the space of polynomials on "V""n" commuting with the action of the orthogonal group.