The second fundamental theorem of invariant theory for the orthogonal group

Let V = C-n be endowed with an orthogonal form and G = O(V) be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism nu : B-r(n) -> End(G)(V-circle times r), where B-r(n) is the r-string Brauer algebra with parameter n. However the kernel of nu has remained elusive. In this paper we show that, in analogy with the case of GL(V), for r >= n+1, nu has kernel which is generated by a single idempotent element E, and we give a simple explicit formula for E. Using the theory of cellular algebras, we show how E may be used to determine the multiplicities of the irreducible representations of O(V) in V-circle times r. We also show how our results extend to the case where C is replaced by an appropriate field of positive characteristic, and comment on quantum analogues of our results.