THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOGONAL GROUP

By G. Lehrer, Ruibin Zhang. 2011

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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 : Br(n)! EndG(V r ), where Br(n) is the r-string Brauer algebra with parameter n. However the kernel of has remained elusive. In this paper we show that, in analogy with the case of GL(V ), for r n + 1, 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 eld of positive characteristic, and we comment on quantum analogues of our results.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7071978cfbcf24a07bf825d5e0b7b71bf7def933

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Citations in the corpus

Gradings on walled Brauer algebras and Khovanov’s arc algebra (2011, result): “Ψm,n r,s , showing it is generated by an idempotent; see Theorem 8.15. This result is an analogue for the walled Brauer algebra of a result about Brauer algebras obtained recently by Lehrer and Zhang [LZ]. For m,n>0, the ideal kerΨm,nr,s need not be generated by an idempotent; see Remark 6.17. We”