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…)