Throughout the article, we denote for convenience.
So far we have seen:
- the Frobenius map gives a correspondence between symmetric polynomials in of degree d and representations of ;
- there is a correspondence between symmetric polynomials in and polynomial representations of .
Here we will describe a more direct relationship between representations of and polynomial representations of Recall from earlier, that and act on as follows:
and the two actions commute, so as endomorphisms of
Lemma. The subspace of all elements fixed by every is spanned by
Use induction on d; the case d=1 is trivial so suppose d>1. For integers , consider the binomial expansion in :
We claim: for large k, the matrix with (i, j)-entry (where ) has rank d+1.
- Indeed, otherwise there are , not all zero, such that for all large k, which is absurd since this is a polynomial in k.
Hence, we can find a linear combination summing up to:
Thus lies in the subspace spanned by all . By induction hypothesis, the set of all spans the whole space. Hence, the set of all spans . ♦
Proposition. If is an -equivariant map, then it is a linear combination of the image of .
Note that since we have Hence from the given condition
By the above lemma, f is a linear combination of for all Since is dense, f is also a linear combination of for . ♦
Now let U be any complex vector space and consider the complex algebra Recall: if is any subset,
is called the centralizer of A. Clearly is a subalgebra and we have
Theorem (Schur-Weyl Duality). Let be a subalgebra which is semisimple. Then:
- is semisimple;
- ; (double centralizer theorem)
- U decomposes as , where are respectively complete lists of irreducible A-modules and B-modules.
Since A is semisimple, we can write it as a finite product . Each simple A-module is of the form for some As an A-module, we can decompose: Here since as A-modules we have:
By Schur’s lemma if and 0 otherwise. This gives:
which is also semisimple. Now each simple B-module has dimension . From the action of B on U, we can write where A acts on the and B acts on the . Expressed as a sum of simple B-modules, we have ; thus repeating the above with A replaced by B gives:
From we thus have This proves all three properties. ♦
From the proof, we see that
- as complex vector spaces,
- acts on the , and
- acts on the .
Thus the correspondence between and works as follows:
The nice thing about this point-of-view is that the construction is now functorial, i.e. for any A-module M, we can define the corresponding: This functor is additive, i.e. , since the Hom functor is bi-additive.
The Case of Sd and GLnC
Now for our main application.
Consider and acting on ; their actions span subalgebras . Now A is semisimple since it is a quotient of . From the lemma, we have B = C(A) so Schur-Weyl duality says A = C(B), B is semisimple and
where are complete lists of simple A– and B-modules respectively. Since A is a quotient of , the are also irreps of so they can be parametrized by .
Proposition. If is the irrep for isomorphic to , then is the irrep for corresponding to
It suffices to show: if corresponds to via the functor in the above note, then
By definition Recall that is a transitive -set; picking a point , any map which is -equivariant is uniquely defined by the element , as long as this element is invariant under the stabilizer group:
Thus, the coefficients of in remain invariant when acted upon by . So we have an element of ♦
Theorem. The set of irreps of occurring in is:
The following is the complete set of -irreps of degree d:
We claim that this is also the set of all irreps in Clearly, each irrep in is of degree d; conversely, has
Clearly so contains all of degree d. Now apply the above proposition. ♦
The simplest non-trivial example follows from the decomposition
The action of is trivial on the first component and alternating on the second.