Appendix F Decomposition of Tensor Spaces for O ( V ) and

In this appendix we complete the results of Chapter 10 concerning the decomposition of tensor space under the action of the orthogonal group or the symplectic group. Following ideas of Weyl [7], we decompose the full tensor space into spaces of partially harmonic tensors. This is the full tensor generalization of the decompositions of the algebras of symmetric tensors and skew-symmetric tensors obtained in Chapter 5. F.1 Partially harmonic tensors Let V be a finite-dimensional complex vector space, and let G ⊂ GL(V ) be the group preserving the nondegenerate symmetric or skew-symmetric bilinear form ω on V . We consider the problem of decomposing the tensor space ⊗k V under the joint action of G and the centralizer algebra Bk(V,ω) = EndG( ⊗k V ) (following the notation of Section 10.1.1). For r = 0,1, . . . , [k/2] let Bk,r(V,ω) be the subspace of Bk(V,ω) spanned by the operators involving r or more contractions (these operators correspond to Brauer diagrams with r or more bars). From the relations in Section 10.1.2 we see that Bk,r(V,ω) is a 2-sided ideal in Bk(V,ω). Thus we have a chain of ideals Bk(V,ω) = Bk,0(V,ω)⊃Bk,1(V,ω)⊃ ·· · ⊃Bk, [k/2](V,ω) . We set T⊗k r = Bk,r(V,ω)( ⊗k V ) (we will not indicate the dependence of these spaces on the pair (V,ω), which will remain fixed throughout this section). Clearly, T⊗k r ⊃ T⊗k r+1, and we have a filtration ⊗kV = T⊗k 0 ⊃ T⊗k 1 ⊃ ·· · ⊃ T⊗k [k/2] (F.1)