Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers

Every elementary tensor expands as $z\otimes v={\sum }_{s}zv(s)(1\otimes e_s)$, so the classes $1\otimes e_s$ span. For each $t\in S$, the balanced map $(z,v)\mapsto zv(t)$ induces by [F6] a map $p_t:V_{\mathbb{C}}\to \mathbb{C}$; it is complex-linear on elementary tensors and satisfies $p_t(1\otimes e_s)={\delta }_{ts}$. Applying $p_t$ to a linear relation proves independence. Thus these classes form a $\mathbb{C}$-basis and ${\dim }_{\mathbb{C}}{V}_{\mathbb{C}}=n$. For $w\in W$ set ${\rho }_{\mathbb{C}}(w):={\mathrm{i}\mathrm{d}}_{\mathbb{C}}\otimes \rho (w)$. By [F7] each ${\rho }_{\mathbb{C}}(w)$ is $\mathbb{C}$-linear with ${\rho }_{\mathbb{C}}(w)(z\otimes v)=z\otimes \rho (w)v$; on elementary tensors ${\rho }_{\mathbb{C}}(w){\rho }_{\mathbb{C}}(w^{\mathrm{\prime }})(z\otimes v)=z\otimes \rho (w)\rho (w^{\mathrm{\prime }})v={\rho }_{\mathbb{C}}(w{w}^{\mathrm{\prime }})(z\otimes v)$, and since elementary tensors span $V_{\mathbb{C}}$ this gives ${\rho }_{\mathbb{C}}(w{w}^{\mathrm{\prime }})={\rho }_{\mathbb{C}}(w){\rho }_{\mathbb{C}}(w^{\mathrm{\prime }})$, while ${\rho }_{\mathbb{C}}(1)=\mathrm{i}\mathrm{d}$; so ${\rho }_{\mathbb{C}}:W\to \mathrm{G}\mathrm{L}(V_{\mathbb{C}})$ is a homomorphism. If ${\rho }_{\mathbb{C}}(w)={\mathrm{i}\mathrm{d}}_{V_{\mathbb{C}}}$, then for every $s$ the basis element $1\otimes e_s$ is fixed, so $1\otimes \rho (w)e_s=1\otimes e_s$, and linear independence of the $1\otimes e_t$ gives $\rho (w)e_s=e_s$; as $(e_s)$ spans $V$, $\rho (w)={\mathrm{i}\mathrm{d}}_V$, and injectivity of $\rho$ [F3] gives $w=1$. Hence ${\rho }_{\mathbb{C}}$ is faithful in the sense of [F8], and its image, the image of the finite group $W$, is a finite subgroup of $\mathrm{G}\mathrm{L}(V_{\mathbb{C}})$ of order $\mathrm{\mid }W\mathrm{\mid }$.

Let $s\in S$. By [F1], $B(e_s,e_s)=1$ and $r_s(v)=v-2B(v,e_s)e_s$ for all $v\in V$; equivalently $r_s-{\mathrm{i}\mathrm{d}}_V=-2B(-,e_s)e_s$ has image $\mathbb{R}{e}_s$ and kernel $\ker B(-,e_s)$, a hyperplane of $V$. Complexifying with [F7], for $x=z\otimes v$ we get $({\rho }_{\mathbb{C}}(s)-{\mathrm{i}\mathrm{d}}_{V_{\mathbb{C}}})(x)=z\otimes (r_s-{\mathrm{i}\mathrm{d}}_V)v=-2B(v,e_s)(z\otimes e_s)=-2B_{\mathbb{C}}(x,e_s)e_s$, using $B_{\mathbb{C}}(z\otimes v,e_s)=zB(v,e_s)$; the same formula holds on all of $V_{\mathbb{C}}$ by linearity. The image is therefore the line $\mathbb{C}{e}_s$ (it contains $-2e_s\mathrm{\ne }0$) and the kernel is $\{x:B_{\mathbb{C}}(x,e_s)=0\}$, a complex hyperplane because $B_{\mathbb{C}}(e_s,e_s)=1$ makes the linear form $B_{\mathbb{C}}(-,e_s)$ nonzero. In particular ${\rho }_{\mathbb{C}}(s)\mathrm{\ne }{\mathrm{i}\mathrm{d}}_{V_{\mathbb{C}}}$ fixes that hyperplane pointwise: each generator is a complex reflection, and ${\rho }_{\mathbb{C}}(W)=⟨{\rho }_{\mathbb{C}}(s):s\in S⟩$ is generated by complex reflections.

Let $t\in T$. By [F5] there is a unique $\alpha \in {\mathrm{\Phi }}_{+}$ with $t=t_{\alpha }$ and $\rho (t_{\alpha })=r_{\alpha }$; by [F3] every root has $B$-norm one, so $r_{\alpha }(x)=x-2B(x,\alpha )\alpha$ for $x\in V$, and $r_{\alpha }$ fixes $\ker B(-,\alpha )$ pointwise and maps $\alpha \mapsto -\alpha$. Complexifying as in 2.1 gives $({\rho }_{\mathbb{C}}(t_{\alpha })-\mathrm{i}\mathrm{d})(x)=-2B_{\mathbb{C}}(x,\alpha )\alpha$ for all $x\in V_{\mathbb{C}}$, so the image is the line $\mathbb{C}\alpha$ and, since $B_{\mathbb{C}}(\alpha ,\alpha )=1$, the kernel $\{x:{\mathrm{\ell }}_{\alpha }(x)=0\}$ is a complex hyperplane. Thus every $t\in T$ acts as a complex reflection, and by [F2] and [F3] every element of $T$ is conjugate in $W$ to a simple reflection, in agreement with 2.1.

Let $x={\sum }_{s\in S}{z}_s\otimes e_s\in V_{\mathbb{C}}$ be fixed by every ${\rho }_{\mathbb{C}}(w)$. Fixing $x$ under ${\rho }_{\mathbb{C}}(s)$ and applying the formula of 2.1 gives $-2B_{\mathbb{C}}(x,e_s)e_s=0$, hence $B_{\mathbb{C}}(x,e_s)=0$ for every $s\in S$. Let $G$ be the Gram matrix of $B$ in the basis $(e_s)$; by [F4] $B$ is positive definite, so $G$ is a real symmetric positive definite matrix, and $B_{\mathbb{C}}(x,e_s)=0$ for all $s$ says $Gz=0$ for the coefficient column $z=(z_s)$. Writing $z=a+bi$ with real columns $a,b$, the real part $a^{T}Ga+b^{T}Gb$ of $z^{\ast }Gz$ vanishes, and both summands are $\ge 0$ with equality only for $a=b=0$; hence $z=0$ and $x=0$. So $V_{\mathbb{C}}^W=\{0\}$. If $\mathrm{\ell }:V_{\mathbb{C}}\to \mathbb{C}$ is a $W$-invariant linear form, then $\mathrm{\ell }(e_s)=\mathrm{\ell }({\rho }_{\mathbb{C}}(s)e_s)=\mathrm{\ell }(-e_s)$ by 2.1, so $\mathrm{\ell }(e_s)=0$ for every $s$, and since the $e_s$ span $V_{\mathbb{C}}$ over $\mathbb{C}$, $\mathrm{\ell }=0$. These finite-dimensional calculations use no choice principle: the basis $(e_s)$, the Gram matrix and the coefficient column are explicitly given by the finite set $S$.

Assume AC. By step 1.1 the subgroup ${\rho }_{\mathbb{C}}(W)\le \mathrm{G}\mathrm{L}(V_{\mathbb{C}})$ is finite of order $\mathrm{\mid }W\mathrm{\mid }$, faithful and, by steps 2.1 and 3.1, generated by complex reflections; $V_{\mathbb{C}}$ has dimension $n$. Applying [F11] with $S=\mathbb{C}[V_{\mathbb{C}}]$ and $G={\rho }_{\mathbb{C}}(W)$ gives that $S^W$ is a polynomial $\mathbb{C}$-algebra on $n$ homogeneous algebraically independent basic invariants, that $S$ is graded free of rank $\mathrm{\mid }W\mathrm{\mid }$ over $S^W$, that every minimal homogeneous invariant generating family of $I=S{S}_{+}^W$ generates $S^W$ and is algebraically independent, that such generators form a regular sequence with the freeness conclusion, and the stated Hilbert-series and order conclusions for $S\mathrm{/}I$; by [F9] this $I$ is exactly the positive-degree ideal of the coinvariant algebra of the definition. This is the only place where AC is used.

For $n=0$ the group $W$ is trivial, $V=V_{\mathbb{C}}=0$, the basis family is empty, ${\rho }_{\mathbb{C}}$ is the unique map of the trivial groups, $S=\mathbb{C}$ and all products and families displayed in (4) are empty; every assertion above holds in this empty form. Nothing in steps 1.1-4.1 uses irreducibility of the diagram, crystallographic integrality, or highest-root data: only the finiteness of $W$, the positivity of $B$, the bijection ${\mathrm{\Phi }}_{+}\to T$ and the reflection formula are used, so reducible systems and the noncrystallographic types $I_2(m)$, $H_3$, $H_4$ are covered. No claim is made here about eigenvalues of a Coxeter element, about the values of the degrees, or about $S\mathrm{/}I$ as a $W$-representation; and the only clause depending on AC is (4), consumed through [F11].