Finite Weyl positive roots and simple reflections

Choose any finite basis $e_1,\dots ,e_r$ of $E$. For each nonzero root $\alpha$, the polynomial ${\sum }_{j=1}^r(e_j,\alpha )z^{j-1}$ is nonzero, since nondegeneracy and spanning prevent all its coefficients vanishing. A nonzero degree-$d$ real polynomial has at most $d$ roots: division by $z-a$ at a zero and induction prove this assertion. Finitely many such polynomials therefore have only finitely many forbidden real values. Choose one other value $z$ and take $t={\sum }_j{z}^{j-1}{e}_j$. In rank zero take $t=0$. Also $W$ acts on the finite spanning set $\mathrm{\Phi }$ and an operator fixing every root is the identity; this injects $W$ into a finite permutation group. Hence $W$ is finite.

If independent roots $\alpha ,\beta$ have $(\alpha ,\beta )<0$, the positive integers $m=-2(\alpha ,\beta )\mathrm{/}(\alpha ,\alpha )$ and $n=-2(\alpha ,\beta )\mathrm{/}(\beta ,\beta )$ obey $mn<4$, by the strict Cauchy–Schwarz inequality. The latter follows directly by minimizing $(\alpha -u\beta ,\alpha -u\beta )>0$ over real $u$. Thus at least one of $m,n$ equals one. The corresponding root reflection produces $\alpha +\beta$, proving it is a root. If distinct simple positive roots had positive inner product, apply this result to one and the negative of the other. Their difference would be a root; whichever sign it has expresses one of the simple roots as a sum of two positive roots, a contradiction. Distinct simple roots therefore have nonpositive inner products.

Every positive root is a sum of simple roots with nonnegative integer coefficients. If not simple, split it into two positive roots. Both have smaller $t$-value, and induction on the position in the finite ordered list of positive $t$-values terminates the splitting. Thus the simple roots span $E$. They are independent: in a nontrivial real relation separate the positive and negative coefficients to obtain $v={\sum }_{i\in I}{a}_i{\alpha }_i={\sum }_{j\in J}{b}_j{\alpha }_j$ with disjoint index sets and strictly positive coefficients. Neither side can be empty, since pairing with $t$ would then give a contradiction; it also shows $v\mathrm{\ne }0$. Step 1.2 gives $(v,v)={\sum }_{i,j}{a}_i{b}_j({\alpha }_i,{\alpha }_j)\le 0$, contradicting positive definiteness. Hence they are a basis. Negating gives the negative-root coordinates as well.

For $\beta \in {\mathrm{\Phi }}^{+}\setminus \{{\alpha }_i\}$, its simple expansion has a positive coefficient at some index other than $i$: otherwise reducedness would make $\beta ={\alpha }_i$. Reflection $s_i$ changes only the coefficient at $i$. Its image is a root and still has that other positive coefficient, so step 2.1 makes every coefficient nonnegative. Thus $s_i\beta$ is positive. The involution $s_i$ then permutes all these positive roots and reverses ${\alpha }_i$.

The coroot set ${\mathrm{\Phi }}^{\vee }$ is a finite reduced crystallographic root system: $s_{\alpha }{\beta }^{\vee }=(s_{\alpha }\beta {)}^{\vee }$, its double dual is $\mathrm{\Phi }$, and its two integer pairings interchange those of $\mathrm{\Phi }$. Its positive roots are positive scalar multiples of the positive roots of $\mathrm{\Phi }$, and every positive coroot has nonnegative real coordinates in the basis ${\alpha }_i^{\vee }$. Apply step 2.1's simple-basis conclusion to this dual system. The cone spanned by its positive roots is exactly the cone generated by the ${\alpha }_i^{\vee }$, since those coroots themselves belong to it. The extreme rays of a cone spanned by a basis are exactly its basis rays: a vector with two positive coordinates splits into two nonproportional cone vectors, whereas a single-coordinate vector cannot. The dual simple roots must therefore lie on exactly these rays. Reducedness gives that they are precisely the ${\alpha }_i^{\vee }$. Their integer positive expansions give $Q^{\vee }={⨁}_i\mathbb{Z}{\alpha }_i^{\vee }$.

Suppose a positive root $\beta ={\sum }_j{n}_j{\alpha }_j$ is not simple. Since $(\beta ,\beta )={\sum }_j{n}_j(\beta ,{\alpha }_j)>0$, some $(\beta ,{\alpha }_i)>0$. Its positive integral coroot pairing means $s_i\beta =\beta -m{\alpha }_i$ with an integer $m>0$. By step 3.1 this root stays positive and has smaller integer height ${\sum }_j{n}_j$. Repeating finitely many times reaches a simple root. Therefore every positive root is in the group orbit of a simple root; a negative root is obtained by first negating that simple root with its reflection. For any orthogonal $u$, direct substitution gives $s_{u\alpha }=u{s}_{\alpha }{u}^{-1}$. Consequently every root reflection lies in the group generated by the $s_i$, proving that this group is $W$.

Let ${\omega }_i$ be the basis dual to ${\alpha }_i^{\vee }$ under the nondegenerate form. Step 3.2 implies $P={⨁}_i\mathbb{Z}{\omega }_i$. Reflection invariance of the root and coroot sets gives invariance of $Q,Q^{\vee }$. If $\lambda \in P$ and $w\in W$, then $(w\lambda ,{\alpha }^{\vee })=(\lambda ,w^{-1}{\alpha }^{\vee })\in \mathbb{Z}$; hence $W$ preserves $P$. Every basis and root list used was finite. In rank zero all lists are empty, their groups are zero and $W$ is trivial, so the same conclusions hold.