Weyl reflection ($s_{\alpha }$)

The reflection associated with a root is $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$.

The reflection group of a root system $\Phi$ is the subgroup of the orthogonal group generated by the reflections $s_{\alpha }$ for $\alpha \in \Phi$. It permutes the roots and acts freely and transitively on the fundamental systems.

The Weyl group is generated by the reflections in the roots.

For a compact Lie group that may be disconnected, fix a maximal torus $T$ and define the normalizer quotient $N_G(T)/T$. It is finite. To prove this, identify $T$ with its Lie algebra modulo the kernel lattice of its exponential map. A torus automorphism preserves this lattice. Its differential is therefore an integral invertible matrix in a lattice basis. A connected subgroup of the normalizer induces a continuous family of such matrices, hence acts trivially on $T$.

The identity component $N_G(T{)}^0$ thus centralizes $T$ and lies in $G^0$. The maximal-torus centralizer theorem for the connected compact group gives $N_G(T{)}^0=T$. The quotient is consequently discrete and compact, hence finite. Its action on $T$ need not be faithful when $G$ is disconnected: for a finite group, $T$ is trivial and $N_G(T)/T=G$. This is why the quotient $N_G(T)/C_G(T)$ must not silently replace $N_G(T)/T$ in this convention.

The reflection representation is the action of the Weyl group on the real span of its root system, generated by its Weyl reflections. For an irreducible root system, it is an irreducible representation: if $U$ is an invariant subspace, each root either lies in $U^{\perp }$ or, by subtracting a reflected vector of $U$, lies in $U$. These alternatives partition the roots into orthogonal sets. Irreducibility and spanning force $U=0$ or the whole space.

The longest element of a finite Weyl group is the element sending all positive roots of a root system to negative roots. It is the longest element of a finite Coxeter group. For the G2 root system it is $-I$: its twelve roots form two hexagons, and the dihedral reflection group of a root system contains the rotation through $\pi$. It determines the highest weight of a dual representation.

The Coxeter length $\ell (w)$ is the smallest number of simple reflections whose product is $w$. Its parity gives the sign of a Weyl-group element.

This is a particular length function, namely word length for the simple reflections.

If $\lambda$ and $w\lambda$ both lie in the closed dominant chamber, then $w$ is a product of simple reflections whose walls contain $\lambda$. In particular, $w\lambda =\lambda$.

For a chosen positive system of a root system $\Pi$, the inversion set isFor a finite Weyl group, $|N(w)|=\ell (w)$.

Its cardinality is the Coxeter length; this root-sign definition should not be silently identified with an unspecified permutation inversion convention.

In the geometric representation of a Coxeter system,andReplacing “positive” by “negative” reverses either length inequality.