Weyl group

A key result about the Weyl group is this:[4]

Theorem: If ${\displaystyle \Delta }$ is base for ${\displaystyle \Phi }$, then the Weyl group is generated by the reflections ${\displaystyle s_{\alpha }}$ with ${\displaystyle \alpha }$ in ${\displaystyle \Delta }$.

That is to say, the group generated by the reflections ${\displaystyle s_{\alpha },\,\alpha \in \Delta ,}$ is the same as the group generated by the reflections ${\displaystyle s_{\alpha },\,\alpha \in \Phi }$.

Meanwhile, if ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are in ${\displaystyle \Delta }$, then the Dynkin diagram for ${\displaystyle \Phi }$ relative to the base ${\displaystyle \Delta }$ tells us something about how the pair ${\displaystyle \{s_{\alpha },s_{\beta }\}}$ behaves. Specifically, suppose ${\displaystyle v}$ and ${\displaystyle v'}$ are the corresponding vertices in the Dynkin diagram. Then we have the following results:

• If there is no bond between ${\displaystyle v}$ and ${\displaystyle v'}$, then ${\displaystyle s_{\alpha }}$ and ${\displaystyle s_{\beta }}$ commute. Since ${\displaystyle s_{\alpha }}$ and ${\displaystyle s_{\beta }}$ each have order two, this is equivalent to saying that ${\displaystyle (s_{\alpha }s_{\beta })^{2}=1}$.
• If there is one bond between ${\displaystyle v}$ and ${\displaystyle v'}$, then ${\displaystyle (s_{\alpha }s_{\beta })^{3}=1}$.
• If there are two bonds between ${\displaystyle v}$ and ${\displaystyle v'}$, then ${\displaystyle (s_{\alpha }s_{\beta })^{4}=1}$.
• If there are three bonds between ${\displaystyle v}$ and ${\displaystyle v'}$, then ${\displaystyle (s_{\alpha }s_{\beta })^{6}=1}$.

The preceding claim is not hard to verify, if we simply remember what the Dynkin diagram tells us about the angle between each pair of roots. If, for example, there is no bond between the two vertices, then ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are orthogonal, from which it follows easily that the corresponding reflections commute. More generally, the number of bonds determines the angle ${\displaystyle \theta }$ between the roots. The product of the two reflections is then a rotation by angle ${\displaystyle 2\theta }$ in the plane spanned by ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, as the reader may verify, from which the above claim follows easily.

Weyl groups are examples of finite reflection groups, as they are generated by reflections; the abstract groups (not considered as subgroups of a linear group) are accordingly finite Coxeter groups, which allows them to be classified by their Coxeter–Dynkin diagram. Being a Coxeter group means that a Weyl group has a special kind of presentation in which each generator x_i is of order two, and the relations other than x_i²=1 are of the form (x_ix_j)^m_ij=1. The generators are the reflections given by simple roots, and m_ij is 2, 3, 4, or 6 depending on whether roots i and j make an angle of 90, 120, 135, or 150 degrees, i.e., whether in the Dynkin diagram they are unconnected, connected by a simple edge, connected by a double edge, or connected by a triple edge. We have already noted these relations in the bullet points above, but to say that ${\displaystyle W}$ is a Coxeter group, we are saying that those are the only relations in ${\displaystyle W}$.

Weyl groups have a Bruhat order and length function in terms of this presentation: the length of a Weyl group element is the length of the shortest word representing that element in terms of these standard generators. There is a unique longest element of a Coxeter group, which is opposite to the identity in the Bruhat order.

Let ${\displaystyle K}$ be a connected compact Lie group and let ${\displaystyle T}$ be a maximal torus in ${\displaystyle K}$. We then introduce the normalizer of ${\displaystyle T}$ in ${\displaystyle K}$, denoted ${\displaystyle N(T)}$ and defined as

${\displaystyle N(T)=\{x\in K|xtx^{-1}\in T,\,{\text{for all }}t\in T\}}$.

We also define the centralizer of ${\displaystyle T}$ in ${\displaystyle K}$, denoted ${\displaystyle Z(T)}$ and defined as

${\displaystyle Z(T)=\{x\in K|xtx^{-1}=t\,{\text{for all }}t\in T\}}$.

The Weyl group ${\displaystyle W}$ of ${\displaystyle K}$ (relative to the given maximal torus ${\displaystyle T}$) is then defined initially as

${\displaystyle W=N(T)/T}$.

Eventually, one proves that ${\displaystyle Z(T)=T}$,[6] at which point one has an alternative description of the Weyl group as

${\displaystyle W=N(T)/Z(T)}$.

Now, one can define a root system ${\displaystyle \Phi }$ associated to the pair ${\displaystyle (K,T)}$; the roots are the nonzero weights of the adjoint action of ${\displaystyle T}$ on the Lie algebra of ${\displaystyle K}$. For each ${\displaystyle \alpha \in \Phi }$, one can construct an element ${\displaystyle x_{\alpha }}$ of ${\displaystyle N(T)}$ whose action on ${\displaystyle T}$ has the form of reflection.[7] With a bit more effort, one can show that these reflections generate all of ${\displaystyle N(T)/Z(T)}$.[6] Thus, in the end, the Weyl group as defined as ${\displaystyle N(T)/T}$ or ${\displaystyle N(T)/Z(T)}$ is isomorphic to the Weyl group of the root system ${\displaystyle \Phi }$.

For a complex semisimple Lie algebra, the Weyl group is simply defined as the reflection group generated by reflections in the roots – the specific realization of the root system depending on a choice of Cartan subalgebra.

For a Lie group G satisfying certain conditions,[note 1] given a torus T < G (which need not be maximal), the Weyl group with respect to that torus is defined as the quotient of the normalizer of the torus N = N(T) = N_G(T) by the centralizer of the torus Z = Z(T) = Z_G(T),

${\displaystyle W(T,G):=N(T)/Z(T).\ }$

The group W is finite – Z is of finite index in N. If T = T₀ is a maximal torus (so it equals its own centralizer: ${\displaystyle Z(T_{0})=T_{0}}$) then the resulting quotient N/Z = N/T is called the Weyl group of G, and denoted W(G). Note that the specific quotient set depends on a choice of maximal torus, but the resulting groups are all isomorphic (by an inner automorphism of G), since maximal tori are conjugate.

If G is compact and connected, and T is a maximal torus, then the Weyl group of G is isomorphic to the Weyl group of its Lie algebra, as discussed above.

For example, for the general linear group GL, a maximal torus is the subgroup D of invertible diagonal matrices, whose normalizer is the generalized permutation matrices (matrices in the form of permutation matrices, but with any non-zero numbers in place of the '1's), and whose Weyl group is the symmetric group. In this case the quotient map N → N/T splits (via the permutation matrices), so the normalizer N is a semidirect product of the torus and the Weyl group, and the Weyl group can be expressed as a subgroup of G. In general this is not always the case – the quotient does not always split, the normalizer N is not always the semidirect product of W and Z, and the Weyl group cannot always be realized as a subgroup of G.[5]