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The second fundamental theorem of invariant theory for the orthogonal group. (English) Zbl 1263.20043

Let \(V\) be a complex, finite-dimensional vector space of dimension \(n\) with an orthogonal form and let \(O(V)\) be the corresponding orthogonal group. There is an associated algebra \(B_r(n)\) called the \(r\)-string Brauer algebra, and a surjective homomorphism \(\nu\colon B_r(n)\to\text{End}_{O(V)}(V^{\otimes r})\) by a result of R. Brauer [Ann. Math. (2) 38, 857-872 (1937; Zbl 0017.39105)]. In the present paper, the authors show that the kernel of \(\nu\) is generated by a single idempotent element \(E\in B_r(n)\) which they describe explicitly.

MSC:

20G05 Representation theory for linear algebraic groups
14L24 Geometric invariant theory
20C08 Hecke algebras and their representations
16G20 Representations of quivers and partially ordered sets
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