On the structure of Brauer's centralizer algebras (Q1109893)

Let G be a group of linear transformations of a vector space V over a field k of characteristic 0, and \(V^ f\) the f-th tensor power of V. The question is how does \(V^ f\) decompose into irreducible representations of G. This is closely related with the structure of the algebra \(B_ f(G)\) of centralizers. For example, \(B_ f(GL(n))\) is a quotient of the group algebra \(kS_ f\) of the symmetric group, which allows to obtain the decomposition of \(V^ f\) in this case. In the paper, algebras \(D_ f(n)\) are studied which play for other classical groups a role similar to that of \(kS_ f\) for GL(n). More precisely, if G is O(n) or Sp(2n), the corresponding algebras \(B_ f(G)\) are quotients of Brauer's \(D_ f(n)\) and \(D_ f(-2n)\), respectively. The algebra \(D_ f(n)\) was studied by various authors mainly using combinatorial methods. P. Hanlon and D. Wales conjectured that this algebra is semisimple for all integers \(f>1\) when n is not an integer. This is proved in the paper. Moreover, when n is an integer, the structure of the semisimple quotient of \(D_ f(n)\) is determined. The main tools go back to the work of V. Jones on subfactors of von Neumann factors and link invariants.