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

Proposition 1 (Aronhold method) There is a natural {G}-isomorphism between homogeneous polynomial functions of degree {m} on {V} and symmetric, multilinear maps {V \times \dots \times V \rightarrow k} (where there are {m} factors).

Proof: It is clear that, given a multilinear, symmetric map {g: V \times \dots \times V \rightarrow k}, we can get a homogeneous polynomial of degree {m} on {V} via {v \mapsto g(v, v, \dots, v)} 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 {V \times \dots \times V \rightarrow k} are the same thing as symmetric {k}-linear maps {V^{\otimes m} \rightarrow k}; these are naturally identified with maps {\mathrm{Sym}^m V \rightarrow k}. So what this proposition amounts to saying is that we have a natural isomorphism

\displaystyle \mathrm{Sym}^m V^* \simeq (\mathrm{Sym}^m V)^*.

But this is eminently reasonable, since there is a functorial isomorphism {(V^{\otimes m})^* \simeq (V^{*})^{\otimes m}} 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 {G}-action on {V}, one can check that the polarization and diagonal imbeddings are {G}-equivariant. \Box

2. Schur-Weyl duality

Let {V} be a vector space. Now we take {G = GL(V)} acting on a tensor power {V^{\otimes m}}; this is the {m}th tensor power of the tautological representation on {V}. However, we have on {V^{\otimes m}} not only the natural action of {GL(V)}, but also the action of {S_m}, given by permuting the factors. These in fact commute with each other, since {GL(V)} acts by operators of the form { A\otimes A \otimes \dots \otimes A} and {S_m} acts by permuting the factors.

Now the representations of these two groups {GL(V)} and {S_m} on {V^{\otimes m}} are both semisimple. For {S_m}, it is because the group is finite, and we can invoke Maschke’s theorem. For {GL(V)}, 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 2 Let {A \subset \mathrm{End}(V^{\otimes m})} be the algebra generated by {GL(V)}, and let {B \subset \mathrm{End}(V^{\otimes m})} be the subalgebra generated by {S_m}. Then {A, B} are the centralizers of each other in the endomorphism algebra. (more…)