Tensor Representations of O(V) and Sp(V)

Abstract

In this chapter we analyze the action of the orthogonal and symplectic groups on the tensor powers of their defining representations. We show (following ideas of Weyl [164]) that the subspaces of harmonic tensors can be decomposed using the theory of Young symmetrizers from Chapter 9. This furnishes models (Weyl modules) for all the irreducible representations of the orthogonal and symplectic groups as spaces of harmonic tensors in the image of Young symmetrizers. Our approach involves the interplay of the commuting algebra (a quotient of the Brauer algebra) with the representation theory of the orthogonal and symplectic groups. The key observation is that the action of the Brauer algebra on the space of harmonic tensors factors through the action of the symmetric group on tensors.

The Riemannian curvature tensor of a pseudo-Riemannian manifold plays a central role in differential geometry, Lie groups, and physics (through Einstein’s theory of general relativity). We use the results of Chapters 9 and the present chapter to analyze the symmetry properties of curvature tensors. We show that the space of all curvature tensors at a fixed point of a manifold is irreducible under the action of the general linear group. Under the orthogonal group, this space decomposes into irreducible subspaces corresponding to scalar curvature, traceless Ricci curvature, and Weyl conformal curvature parts. We determine these subspaces using earlier results in this chapter together with the theorem of the highest weight and the Weyl dimension formula. In the last section of the chapter we apply representation theory to knot theory. We use the invariant theory of the orthogonal group to prove the existence of the Jones polynomial (an invariant of oriented links under ambient oriented diffeomorphism).