With the semester about to start, I have been trying to catch up on more classical material. In this post, I’d like to discuss a foundational result on the ring of invariants of the general linear group acting on polynomial rings: that is, a description of generators for the ring of invariants.

**1. The Aronhold method**

Let be a group acting on a finite-dimensional vector space over an algebraically closed field of characteristic zero. We are interested in studying the invariants of the ring of *polynomial functions* on . That is, we consider the algebra , which has a natural -action, and the subalgebra . Clearly, we can reduce to considering *homogeneous* polynomials, because the action of on polynomials preserves degree.

Proposition 1 (Aronhold method)There is a natural -isomorphism between homogeneous polynomial functions of degree on and symmetric, multilinear maps (where there are factors).

*Proof:* It is clear that, given a multilinear, symmetric map , we can get a homogeneous polynomial of degree on via by the diagonal imbedding. The inverse operation is called *polarization.* I don’t much feel like writing out, so here’s a hand-wavy argument.

Or we can think of it more functorially. Symmetric, multilinear maps are the same thing as symmetric *-linear* maps ; these are naturally identified with maps . So what this proposition amounts to saying is that we have a natural isomorphism

But this is eminently reasonable, since there is a functorial isomorphism functorially, and replacing with the symmetric algebra can be interpreted either as taking invariants or coinvariants for the symmetric group action. Now, if we are given the -action on , one can check that the polarization and diagonal imbeddings are -equivariant.

**2. Schur-Weyl duality**

Let be a vector space. Now we take acting on a tensor power ; this is the th tensor power of the tautological representation on . However, we have on not only the natural action of , but also the action of , given by permuting the factors. These in fact commute with each other, since acts by operators of the form and acts by permuting the factors.

Now the representations of these two groups and on are both semisimple. For , it is because the group is finite, and we can invoke Maschke’s theorem. For , it is because the group is *reductive, *although we won’t need this fact*.* In fact, the two representations are complementary to each other in some sense.

Proposition 2Let be the algebra generated by , and let be the subalgebra generated by . Then are the centralizers of each other in the endomorphism algebra. (more…)