Exercise 3, § 5

Let $K$ be a field of characteristic zero, $V$ a vector space of finite dimension $l$ over $K$, and $G$ a subgroup of $GL(V)$ generated by pseudo-reflections. Put $q = \mathrm{Card}(G)$; denote by $S$ (resp. $L$) the symmetric algebra (resp. exterior algebra) of $V$. If $x \in V$, denote by $x$ (resp. $x'$) its canonical image in $S$ (resp. $L$).

a) Let $E = S \otimes L$ be the tensor product of $S$ and $L$. Show that there exists a unique derivation $d$ on $E$ such that $dx = x'$ and $dx' = 0$ for all $x \in V$.

b) Let $S^G$ be the algebra of symmetric invariants of $G$ and let $P_1, \ldots, P_l$ be homogeneous elements of $S^G$ such that $S^G = K[P_1, \ldots, P_l]$. For every subset $I = \{i_1, \ldots, i_r\}$ of $[1, l]$ with $i_1 < \cdots < i_r$, put

Show that the $\omega_I$ are linearly independent over $S$, and that they belong to the subalgebra $E^G$ consisting of the elements of $E$ invariant under $G$. Deduce that, for all $\omega \in E^G$, there exist $a, c_I \in S^G$, with $a \neq 0$, such that

c) Show that every element of $E^G$ contained in $S \otimes \bigwedge^l V$ is of the form $c.dP_1 \ldots dP_l$, with $c \in S^G$. (Apply Prop. 5.)

d) Show that the $\omega_I$ form a basis of the $S^G$-module $E^G$. (With the notation in b), multiply both sides of the relation $a \omega = \sum_I c_I \omega_I$ by an element $\omega_J$, and apply c); deduce that, if $I$ is the complement of $J$, $a$ divides $c_I$; hence$^9$ the fact that the $\omega_I$ generate $E^G$.)

e) Let $S_n$ (resp. $L_m$) be the homogeneous component of $S$ (resp. $L$) of degree $n$ (resp. $m$). Put

By using d), prove the formula where $p_i = \deg(P_i)$.

(9) For more details, see L. SOLOMON, Invariants of finite reflection groups, Nagoya Math. Journal, v. XXII (1963), p. 57-64.

f) If $g \in G$, let $\mathrm{Tr}_{n,m}(g)$ be the trace of the automorphism of $E_{n,m}$ defined by $g$; put

Show that g) Let $q = \mathrm{Card}(G)$. Show that i.e. (Use the following result: if $G$ acts on a finite dimensional vector space $E$, the dimension of the space of elements of $E$ invariant under $G$ is equal to $\frac{1}{q} \sum_{g \in G} \mathrm{Tr}_E(g)$.) What does this formula give for $Y = 0$? h) For any integer $p \geq 0$, let $H_p$ be the set of elements $g \in G$ that have 1 as an eigenvalue with multiplicity $p$. Let $h_p = \mathrm{Card}(H_p)$. Prove the formula (In the formula in g) above, replace $Y$ by $-1 + T(1 - X)$ and then put $X = 1$ in the result. If $g \in H_p$, the term $\mathrm{Tr}(X, Y)(g)$ becomes $T^p$.)