Exercise 3, § 2

Let V be a complex Hilbert space with basis $e_1, \ldots, e_l$. For $1 \leq i \leq l$, let $s_i$ be a unitary pseudo-reflection, with vector $e_i$, such that $s_i(e_i) = c_i e_i$, with $c_i \neq 1$; an element of V is invariant under $s_i$ if and only if it is orthogonal to $e_i$. Let W be the subgroup of $\mathbf{GL}(V)$ generated by the $s_i$.

a) Let $i$ be an integer $\geq 1$. Show that every element of $\bigwedge^i V$ invariant under W is zero. (Argue by induction on $l$; if $V'$ is the subspace of V generated by $e_1, \ldots, e_{l-1}$, and if $e$ is a non-zero vector orthogonal to $V'$, any element of $\bigwedge^i V$ can be written in the form $a + (b \wedge e)$, with $a \in \bigwedge^i V'$ and $b \in \bigwedge^{i-1} V'$; if $a + (b \wedge e)$ is invariant under W, $a$ and $b$ are invariant under the subgroup $W'$ generated by $s_1, \ldots, s_{l-1}$, and the induction hypothesis applies.)

b) Assume that W is finite. Show, by using a), that, for any endomorphism A of V,

Deduce that, for any $A \in \operatorname{End}(V)$, there exists $w \in W$ such that $Aw$ has no non-zero fixed point.

c) Let $\Gamma$ be the graph whose set of vertices is $\{1, l\}$, the arrows being the sets $\{i, j\}$ such that $e_i$ and $e_j$ are not orthogonal. Show that $V$ is a simple $W$-module if and only if $\Gamma$ is connected and non-empty.

d) Assume that $V$ is a simple $W$-module. Show that the $W$-modules $\bigwedge^i V$ ($0 \leq i \leq l$) are simple. (Show that there exists an integer $j$ such that the graph $\Gamma - \{j\}$ is connected. Argue by induction on $l$, and apply the induction hypothesis to the subspace $V'$ generated by the $e_i, i \neq j$.) Show that these modules are pairwise non-isomorphic$^6$.