The second fundamental theorem of invariant theory for the orthogonal group

Gustav Lehrer, Ruibin Zhang
2012 Annals of Mathematics  
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 ν : Br(n) → EndG(V ⊗r ), where Br(n) is the r-string Brauer algebra with parameter n. However the kernel of ν has remained elusive. In this paper we show that, in analogy with the case of GL(V ), for r ≥ n + 1, ν has a kernel which is generated by a single idempotent element E, and we give a simple explicit formula for E. Using the
more » ... or 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 ⊗r . We also show how our results extend to the case where C is replaced by an appropriate field of positive characteristic, and we comment on quantum analogues of our results.
doi:10.4007/annals.2012.176.3.12 fatcat:t4i5klf7bzfcno7vmev3zy4rlm