is indeed a trace. Representations Brauer-Specht modules Brauer-Specht modules are finite-dimensional modules of the Brauer algebra. If is such that is semisimple, they form a complete set of simple modules of . These modules are parametrized by partitions, because they are built from the Specht modules of the symmetric group, which are themselves parametrized by partitions. For with , let be the set of perfect matchings of elements , such that is matched with one of the elements . For any ring , the space is a left -module, where basis elements of act by graph concatenation. (This action can produce matchings that violate the restriction that cannot match with one another: such graphs must be modded out.) Moreover, the space is a right -module. Given a Specht module of , where is a partition of (i.e. ), the corresponding Brauer-Specht module of is A basis of this module is the set of elements , where is such that the lines that end on elements do not cross, and belongs to a basis of . 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. Schur-Weyl duality Let be a euclidean vector space of dimension , and the corresponding orthogonal group. 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 . (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 , the image of inside is the centraliser of inside , and conversely the image of is the centraliser of . The tensor power is therefore both an - and a -module and satisfies where runs over a subset of the partitions such that and , is an irreducible -module, and is a Brauer-Specht module of . 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. If is a negative even integer, the Brauer algebra is related by Schur-Weyl duality to the symplectic group , rather than