Codex @codex  0

For the chosen simple roots $\Delta$, the open fundamental chamber of a root system isEquivalently $\lambda (h_{\alpha })>0$ for every simple coroot. Since all positive roots are nonnegative combinations of simple roots, requiring the inequality for all positive roots gives the same chamber. It is a connected component of the complement of the root reflection hyperplanes; the Weyl group permutes those chambers simply transitively.

The Closed dominant Weyl chamber is obtained by replacing the inequalities by $\ge 0$. It is the cone generated by the fundamental weights. When the fundamental Weyl chamber is taken to include its walls, this closed convention is intended; both conventions determine the same positive system and boundary rays.

For the usual diagonal Cartan subalgebra, write ${ϵ}_i(H)=H_{ii}$. The roots are ${ϵ}_i-{ϵ}_j$ for $i\ne j$. The three diagonal entries of the specified $H_0$ have orderingTherefore the positive roots selected by $\ell (\mu )=\mu (H_0)$ areTheir positive evaluations are respectively $2-\sqrt{2}$, $2\sqrt{2}-1$ and $1+\sqrt{2}$. For this nonstandard ordering the simple roots are ${\alpha }_1={ϵ}_2-{ϵ}_3$ and ${\alpha }_2={ϵ}_3-{ϵ}_1$; the third positive root is their sum. These simple roots differ from the usual choice used in Question 1.

Identify the real weight plane with triples $(u_1,u_2,u_3)$ of sum zero. Pairing with these simple coroots gives the differences $u_2-u_3$ and $u_3-u_1$. Consequently the fundamental chamber of a root system isIts closure replaces the inequalities by weak inequalities. The fundamental weights for this ordering are ${\omega }_1={ϵ}_2$ and ${\omega }_2=-{ϵ}_1$, represented by $(-1/3,2/3,-1/3)$ and $(-2/3,1/3,1/3)$. Their simple-coroot pairings are $(1,0)$ and $(0,1)$, so the two boundary rays areThe sketch uses ${ϵ}_1=(1,0)$, ${ϵ}_2=(-1/2,\sqrt{3}/2)$ and ${ϵ}_3=(-1/2,-\sqrt{3}/2)$ in a Euclidean plane. The shaded sector is the sixty-degree closed chamber; its interior is the open chamber. The ordering vector $H_0$, identified with its dual direction, lies inside it.

Here a reduced root system is crystallographic, as appropriate to a semisimple Lie algebra. It is a finite spanning set $\Phi \subset E\setminus \{0\}$ in a real inner product space of dimension $r$, invariant under each Weyl reflectionwith $2(\beta ,\alpha )/(\alpha ,\alpha )\in \mathbb{Z}$ for every pair of roots and $\Phi \cap \mathbb{R}\alpha =\{\alpha ,-\alpha \}$. A base of a root system $\Delta =({\alpha }_1,\dots ,{\alpha }_r)$ is a basis such that the coefficients of every root are integers that are either all nonnegative or all nonpositive. These define the positive system of a root system. We use the existing Cartan matrix conventionUsing the alternative denominator $({\alpha }_i,{\alpha }_i)$ transposes all the matrices below.

The classification of rank-two root systems gives $A_1\times A_1$, $A_2$, $B_2=C_2$, and $G_2$. In orthonormal coordinates we may choose the following simple roots and Cartan matrices:

Each complete root system consists of these positive roots and their negatives. To see why the list is exhaustive, the off-diagonal Cartan matrix entries of two distinct simple roots are nonpositive integers, and their product is $4{\cos }^2\theta <4$. Thus the product is $0,1,2$ or $3$, corresponding to simple-root angles $\pi /2,2\pi /3,3\pi /4$ or $5\pi /6$. These determine the four systems and their length ratios.

The Weyl group of a rank-two root system is generated by the two simple Weyl reflections. Their product is a rotation of order $m=2,3,4,6$, respectively, so the groups are the dihedral groups of order $2m=4,6,8,12$. The first is also the Klein four-group and the second the symmetric group $S_3$. The root-orthogonal lines cut the plane into $2m$ Weyl chambers, each of angle $\pi /m$. A chosen fundamental chamber of a root system is $(x,{\alpha }_1)>0$, $(x,{\alpha }_2)>0$; its walls are the two root-orthogonal lines. The Weyl group acts simply transitively on these open chambers.

The crystallographic hypothesis matters: if the integer-pairing condition is dropped, reduced noncrystallographic systems of type $I_2(m)$ also occur. They are not additional Lie-algebra root systems.

Let $C(\Delta )$ be the fundamental chamber of a root system determined by $\Delta$. Every positive root is a nonnegative linear combination of the simple roots, so every point in the interior of $C(\Delta )$ pairs strictly positively with every positive root. In particular, the interior meets none of the reflecting hyperplanes belonging to the subsystem ${\Phi }_0$ of roots of maximal length.

The connected set $C(\Delta )$ therefore lies in one chamber of the long-root subsystem. Let ${\Delta }_0$ be the unique fundamental system of a root system defining that chamber. Taking closures gives $C(\Delta )\subseteq C({\Delta }_0)$. Uniqueness follows because the interiors of two distinct chambers are disjoint. Thus there is a unique such ${\Delta }_0$.

Let $C$ be the fundamental chamber of a root system. The chambers $wC$ and $w{s}_{\alpha }C$ are adjacent across the reflecting hyperplane orthogonal to $w(\alpha )$. The chamber $wC$ lies on the side on which $w(\alpha )$ is positive. If $w(\alpha )\in \Pi$, then $C$ lies on that same side, so crossing this wall moves one step farther from $C$; if $w(\alpha )\in -\Pi$, it moves one step nearer. The gallery distance from $C$ to $wC$ is the Coxeter length $\ell (w)$, and adjacent chamber distances differ by one. Consequently