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The second fundamental theorem of invariant theory for the orthogonal group

2012
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Annals of Mathematics
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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

doi:10.4007/annals.2012.176.3.12
fatcat:t4i5klf7bzfcno7vmev3zy4rlm