If is a finite-dimensional complex vector space, then the symmetric group naturally acts on the tensor power by permuting the factors. This action of commutes with the action of , so all permutations are morphisms of -representations. This defines a morphism , and a natural question to ask is whether this map is surjective.
Part of Schur-Weyl duality asserts that the answer is yes. The double commutant theorem plays an important role in the proof and also highlights an important corollary, namely that admits a canonical decomposition
where runs over partitions, are some irreducible representations of , and are the Specht modules, which describe all irreducible representations of . This gives a fundamental relationship between the representation theories of the general linear and symmetric groups; in particular, the assignment can be upgraded to a functor called a Schur functor, generalizing the construction of the exterior and symmetric products.
The proof below is more or less from Etingof’s notes on representation theory (Section 4.18). We will prove four versions of Schur-Weyl duality involving , and (in the special case that is a complex inner product space) .