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We show that the centraliser of the maximal compact subgroup of the realorthogonal or symplectic groups acting on tensors of their standardrepresentation are isomorphic to cyclotomic Brauer algebras. We also show thatfor the symplectic group this cyclotomic Brauer algebra splits into summands ofwalled Brauer algebras.

Compact Schur-Weyl duality: real Lie groups and the cyclotomic Brauer algebra