Symmetry, Representations, and Invariants pp 387-424 | Cite as

# 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 } C^{ n }. 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 } C^{ n } 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.

## Keywords

Irreducible Representation Tensor Representation Weyl Module Standard Tableau Isotypic Component## Preview

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