## The Chevalley-Shephard-Todd theorem

Let $V$ be a finite dimensional vector space over a field $𝔽$. Let $W$ be a finite subgroup of $\mathrm{GL}\left(V\right)$. If ${S\left(V\right)}^{W}$ is a polynomial algebra then $W$ is generated by reflections.

Proof. Let

 $I=⟨f\in {S\left(V\right)}^{W}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}f\left(0\right)=0⟩$
be the ideal in $S\left(V\right)$ generated by polynomials without constant term. Let ${e}_{1},\dots ,{e}_{r}$ be homogeneous generators of $I$ (which exist, by Hilbert).

Step 1. Every $f\in {S\left(V\right)}^{W}is a polynomial in{e}_{1},\dots ,{e}_{r}.$

Proof. The proof is by induction on the degree of $f$. Assume $f$ is homogeneous and $\mathrm{deg}\left(f\right)>0$. Since $f\in I$,

 $f=\sum _{i=1}^{r}{p}_{i}{e}_{i},\phantom{\rule{2em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}{p}_{i}\in S\left(V\right),$
and so $f= 1|W| ∑w∈W wf = ∑ i=1r ( 1|W| ∑w∈W wpi ) ei,$ and since the internal sum has lower degree it can be written as a polynomial in ${e}_{1},\dots ,{e}_{r}$. $\square$

Step 2. $r=\mathrm{dim}\left(V\right)$.

Proof. Let $n=\mathrm{dim}\left(V\right)$, let ${x}_{1},\dots ,{x}_{n}$ be a basis of $V$ and let $ℂ\left({x}_{1},\dots ,{x}_{n}\right)$ be the field of fractions of $S\left(V\right)=ℂ\left[{x}_{1},\dots ,{x}_{n}\right]$. Since ${x}_{i}$ is a root of $mi(t) = ∏w∈W (t-wxi) ∈ S(V)W [t],$ the variable ${x}_{i}$ is algebraic over $ℂ\left({e}_{1},\dots ,{e}_{r}\right)$, the field of fractions of ${S\left(V\right)}^{W}$. Thus $0= trdeg( ℂ(x1,…, xn) ℂ(e1,…, er) ) = trdeg( ℂ(x1,…, xn) ℂ ) = trdeg( ℂ(e1,…, er) ℂ ) =n-r. □$

Step 3. The Jacobian of a map $φ: V ⟶ V x ⟼ (φ(x), …,φ(x)) is Jφ(x) = det( ∂φi ∂xj ).$ If $\phi$ is linear then there are ${\phi }_{ij}\in ℂ$ such that $φi(x) = ∑j=1n φijxj and Jφ=det(φ ij).$ The chain rule is the identity $Jθ∘φ = Jθ(φx) Jφ(x) .$ Let $θ: V ⟶ V x ⟼ (e1(x), …,en(x)) and w: V ⟶ V x ⟼ wx$ for $w\in W$. Then $\theta \circ w=\theta$ and so $Jθ(x) = Jθ∘w(x) = Jθ(wx) Jw(x) = Jθ(wx) det(w) = det(w) (w-1 Jθ)(x) .$ Thus ${J}_{\theta }$ is $W$-alternating and so ${J}_{\theta }$ is divisible by $Δ= ∏ α∈R+ αsα-1 . Since deg(Jθ) =∑i=1n (di-1) =Card(R+) ,$ it follows that ${J}_{\theta }=\lambda \cdot \Delta$ for some $\lambda \in ℂ$. $\square$

Step 4. The polynomials ${e}_{1},\dots ,{e}_{n}$ are algebraically independent if and only if ${J}_{\theta }\ne 0$.

Proof. $⟹$: Assume ${e}_{1},\dots ,{e}_{r}$ are algebraically independent. Since $trdeg( ℂ(x1,…, xn) ℂ(e1,…, er) ) = trdeg( ℂ(x1,…, xn) ℂ ) = trdeg( ℂ(e1,…, er) ℂ ) ≥n-r,$ ${x}_{1},\dots ,{x}_{n}$ are algebraic over $ℂ\left({e}_{1},\dots ,{e}_{r}\right)$ if and only if $0\ge n-r$, that is, if and only if $n=r$.

Let ${m}_{i}\left(t\right)\in {S\left(V\right)}^{W}\left[t\right]$ be the minimal polynomial of ${x}_{i}$ over $ℂ\left({e}_{1},\dots ,{e}_{r}\right)$, the field of fractions of ${S\left(V\right)}^{W}$. Then $∂mi ∂xk = ∑j=1r ∂mi ∂ej ∂ej ∂xk + ∂mi t t ∂xk$ and $0= ∂mi(xi) ∂xk = = ∑j=1r ∂mi ∂ej (xi) ∂ej ∂xk + mi′(xi) δik .$ Thus $det( ∂mi ∂ej (xi) ) ⋅Jθ = det(-diag( m′1 (x1), … m′n (xn)) ) = (-1)n ∏i=1r m′i (xi) .$ Since ${m}_{i}\left(t\right)$ is the minimal polynomial of ${x}_{i}$, each factor ${m\prime }_{i}\left({x}_{i}\right)\ne 0$ and, thus, ${J}_{\theta }\ne 0$.

$⟸$: Assume ${e}_{1},\dots ,{e}_{n}$ are algebraically independent. Let $f\left({y}_{1},\dots ,{y}_{n}\right)$ be of minimal degree such that $f\left({e}_{1},\dots ,{e}_{n}\right)=0$. Then $∂f ∂yi ≠0 for some yi, and so gi = ∂f ∂yi ( e1,…, en) ≠0 for somei.$ But $0= ∂f(e1,…, en) ∂xj = ∑i=1n ∂f ∂yi (e1,…, en) ∂ei ∂xj , and so ∑i=1n gi ∂ei ∂xj =0.$ So ${g}_{i}$ is a solution to the equation $\left({g}_{1},\dots ,{g}_{n}\right)\left(\partial {e}_{i}/\partial {x}_{j}\right)=0$ and so ${J}_{\theta }=0$. $\square$

## Notes and References

These notes are a retyping of section 3 of http://researchers.ms.unimelb.edu.au/~aram@unimelb/Notespre2005/winvts1.26.04.pdf

## References

[St] R. Steinberg, Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968. iii+277 pp. MR0466335.