Exercise 8, § 5

*Let V be a finite dimensional vector space over a field K, let S be the symmetric algebra of V, and let G be a finite subgroup of $\mathbf{GL}(V)$. Assume

11Cf. M. Auslander, On the purity of the branch locus, Amer. J. of Math., v. LXXXIV (1962), p. 116-125.

that the algebra $S^G$ of symmetric invariants of $G$ is a graded polynomial algebra.

a) Let $u$ be an element of the dual $V^*$ of $V$, and let $G_u$ be the subgroup of $G$ consisting of the elements leaving $u$ invariant. Show that $G_u$ is generated by pseudo-reflections. (The linear form $u$ extends to a homomorphism $f_u : S \to K$. If $S_u$ denotes the local ring of $S$ with respect to the kernel of $f_u$; the ring $S_u$ is regular. Conclude by applying Exerc. 7 b) to $G_u$, considered as a group of automorphisms of $S_u$.) In particular, $G$ is generated by pseudo-reflections.

b) Let $A$ be a subset of $V^*$, and let $G_A = \bigcap_{u \in A} G_u$. Show that $G_A$ is generated by pseudo-reflections. (Extending the base field if necessary, we can assume that $K$ is infinite. Show that in that case there exists an element $v$ of the vector subspace of $V^*$ generated by $A$ such that $G_v = G_A$. Conclude by applying a) to $v$.)