The second fundamental theorem of invariant theory for the orthogonal group | Annals of Mathematics

Abstract Let V=ℂn 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 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.

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数学年刊(Annals of Mathematics)

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