Tensor Representations of GL(V)

Abstract

In this chapter we bring together the representation theories of the groups GL(n,C) and б _k via their mutually commuting actions on ⊗^k Cⁿ. We already exploited this connection in Chapter 5 to obtain the first fundamental theorem of invariant theory for GL(n, C). In this chapter we obtain the full isotypic decomposition of ⊗^k Cⁿ under the action of GL(n,C)× б _k . This decomposition gives the Schur–Weyl duality pairing between the irreducible representations of GL(n,C) and those of б _k . From this pairing we obtain the celebrated Frobenius character formula for the irreducible representations of б _k . We then reexamine Schur–Weyl duality and GL(k,C)–GL(n,C) duality from Chapters 4 and 5 in the framework of dual pairs of reductive groups. Using the notion of seesaw pairs of subgroups, we obtain reciprocity laws for tensor products and induced representations. In particular, we show that every irreducible б _k -module can be realized as the weight space for the character x ↦ det(x) in an irreducible GL(k,C) representation. Explicit models (the Weyl modules) for all the irreducible representations of GL(n,C) are obtained using Young symmetrizers. These elements of the group algebra of C[б _k ] act as projection operators onto GL(n,C)-irreducible invariant subspaces. The chapter concludes with the Littlewood–Richardson rule for calculating the multiplicities in tensor products.