Inmathematics, in particular the theory ofLie algebras, theWeyl group (named afterHermann Weyl) of aroot system Φ is asubgroup of theisometry group of that root system. Specifically, it is the subgroup which is generated by reflections through thehyperplanesorthogonal to at least one of the roots, and as such is afinite reflection group. In fact it turns out thatmost finite reflection groups are Weyl groups.^[1] Abstractly, Weyl groups arefinite Coxeter groups, and are important examples of these.

The Weyl group of asemisimple Lie group, a semisimpleLie algebra, a semisimplelinear algebraic group, etc. is the Weyl group of theroot system of that group or algebra.

Let${\displaystyle \mathrm{\Phi }}$ be aroot system in a Euclidean space${\displaystyle V}$. For each root${\displaystyle \alpha \in \mathrm{\Phi }}$, let${\displaystyle s_{\alpha }}$ denote the reflection about the hyperplane perpendicular to${\displaystyle \alpha }$, which is given explicitly as

${\displaystyle s_{\alpha }(v)=v-2\frac{(v,\alpha )}{(\alpha ,\alpha )}\alpha }$,

where${\displaystyle (\cdot ,\cdot )}$ is the inner product on${\displaystyle V}$. The Weyl group${\displaystyle W}$ of${\displaystyle \mathrm{\Phi }}$ is the subgroup of the orthogonal group${\displaystyle O(V)}$ generated by all the${\displaystyle s_{\alpha }}$'s. By the definition of a root system, each${\displaystyle s_{\alpha }}$ preserves${\displaystyle \mathrm{\Phi }}$, from which it follows that${\displaystyle W}$ is a finite group.

In the case of the${\displaystyle A_2}$ root system, for example, the hyperplanes perpendicular to the roots are just lines, and the Weyl group is the symmetry group of an equilateral triangle, as indicated in the figure. As a group,${\displaystyle W}$ is isomorphic to the permutation group on three elements, which we may think of as the vertices of the triangle. Note that in this case,${\displaystyle W}$ is not the full symmetry group of the root system; a 60-degree rotation preserves${\displaystyle \mathrm{\Phi }}$ but is not an element of${\displaystyle W}$.

We may consider also the${\displaystyle A_n}$ root system. In this case,${\displaystyle V}$ is the space of all vectors in${\displaystyle {\mathbb{R}}^{n+1}}$ whose entries sum to zero. The roots consist of the vectors of the form${\displaystyle e_i-e_j,i\ne j}$, where${\displaystyle e_i}$ is the${\displaystyle i}$th standard basis element for${\displaystyle {\mathbb{R}}^{n+1}}$. The reflection associated to such a root is the transformation of${\displaystyle V}$ obtained by interchanging the${\displaystyle i}$th and${\displaystyle j}$th entries of each vector. The Weyl group for${\displaystyle A_n}$ is then the permutation group on${\displaystyle n+1}$ elements.

If${\displaystyle \mathrm{\Phi }\subset V}$ is a root system, we may consider the hyperplane perpendicular to each root${\displaystyle \alpha }$. Recall that${\displaystyle s_{\alpha }}$ denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of${\displaystyle V}$ generated by all the${\displaystyle s_{\alpha }}$'s. The complement of the set of hyperplanes is disconnected, and each connected component is called aWeyl chamber. If we have fixed a particular set Δ of simple roots, we may define thefundamental Weyl chamber associated to Δ as the set of points${\displaystyle v\in V}$ such that${\displaystyle (\alpha ,v)>0}$ for all${\displaystyle \alpha \in \mathrm{\Delta }}$.

Since the reflections${\displaystyle s_{\alpha },\alpha \in \mathrm{\Phi }}$ preserve${\displaystyle \mathrm{\Phi }}$, they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.

The figure illustrates the case of the A2 root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.

A basic general theorem about Weyl chambers is this:^[2]

Theorem: The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl chambers.

A related result is this one:^[3]

Theorem: Fix a Weyl chamber${\displaystyle C}$. Then for all${\displaystyle v\in V}$, the Weyl-orbit of${\displaystyle v}$ contains exactly one point in the closure${\displaystyle \overline{C}}$ of${\displaystyle C}$.

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

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

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

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

• If there is no bond between${\displaystyle v}$ and${\displaystyle v^{\prime }}$, 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${\displaystyle v^{\prime }}$, then${\displaystyle (s_{\alpha }{s}_{\beta }{)}^3=1}$.
• If there are two bonds between${\displaystyle v}$ and${\displaystyle v^{\prime }}$, then${\displaystyle (s_{\alpha }{s}_{\beta }{)}^4=1}$.
• If there are three bonds between${\displaystyle v}$ and${\displaystyle v^{\prime }}$, 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 accordinglyfinite Coxeter groups, which allows them to be classified by theirCoxeter–Dynkin diagram. Being a Coxeter group means that a Weyl group has a special kind ofpresentation in which each generatorx_i is of order two, and the relations other thanx_i²=1 are of the form (x_ix_j)^m_ij=1. The generators are the reflections given by simple roots, andm_ij is 2, 3, 4, or 6 depending on whether rootsi andj make an angle of 90, 120, 135, or 150 degrees, i.e., whether in theDynkin 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 theonly relations in${\displaystyle W}$.

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

Above, the Weyl group was defined as a subgroup of the isometry group of a root system. There are also various definitions of Weyl groups specific to various group-theoretic and geometric contexts (Lie algebra,Lie group,symmetric space, etc.). For each of these ways of defining Weyl groups, it is a (usually nontrivial) theorem that it is a Weyl group in the sense of the definition at the top of this article, namely the Weyl group of some root system associated with the object. A concrete realization of such a Weyl group usually depends on a choice – e.g. ofCartan subalgebra for a Lie algebra, ofmaximal torus for a Lie group.^[5]

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

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

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

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

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${\displaystyle \mathrm{\Phi }}$ associated to the pair${\displaystyle (K,T)}$; the roots are the nonzeroweights of the adjoint action of${\displaystyle T}$ on the Lie algebra of${\displaystyle K}$. For each${\displaystyle \alpha \in \mathrm{\Phi }}$, 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${\displaystyle \mathrm{\Phi }}$.

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

For aLie groupG satisfying certain conditions,^{[note 1]} given a torusT <G (which need not be maximal), the Weyl groupwith respect to that torus is defined as the quotient of thenormalizer of the torusN =N(T) =N_G(T) by thecentralizer of the torusZ =Z(T) =Z_G(T),

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

The groupW is finite –Z is of finiteindex inN. IfT =T₀ is amaximal torus (so it equals its own centralizer:${\displaystyle Z(T_0)=T_0}$) then the resulting quotientN/Z =N/T is calledtheWeyl group ofG, and denotedW(G). Note that the specific quotient set depends on a choice of maximaltorus, but the resulting groups are all isomorphic (by an inner automorphism ofG), since maximal tori are conjugate.

IfG is compact and connected, andT is amaximal torus, then the Weyl group ofG is isomorphic to the Weyl group of its Lie algebra, as discussed above.

For example, for the general linear groupGL, a maximal torus is the subgroupD of invertible diagonal matrices, whose normalizer is thegeneralized permutation matrices (matrices in the form ofpermutation matrices, but with any non-zero numbers in place of the '1's), and whose Weyl group is thesymmetric group. In this case the quotient mapN →N/T splits (via the permutation matrices), so the normalizerN is asemidirect product of the torus and the Weyl group, and the Weyl group can be expressed as a subgroup ofG. In general this is not always the case – the quotient does not always split, the normalizerN is not always thesemidirect product ofW andZ, and the Weyl group cannot always be realized as a subgroup ofG.^[5]

IfB is aBorel subgroup ofG, i.e., a maximalconnectedsolvable subgroup and a maximal torusT =T₀ is chosen to lie inB, then we obtain theBruhat decomposition

${\displaystyle G=\bigcup _{w\in W}BwB}$

which gives rise to the decomposition of theflag varietyG/B intoSchubert cells (seeGrassmannian).

The structure of theHasse diagram of the group is related geometrically to the cohomology of the manifold (rather, of the real and complex forms of the group), which is constrained byPoincaré duality. Thus algebraic properties of the Weyl group correspond to general topological properties of manifolds. For instance, Poincaré duality gives a pairing between cells in dimensionk and in dimensionn -k (wheren is the dimension of a manifold): the bottom (0) dimensional cell corresponds to the identity element of the Weyl group, and the dual top-dimensional cell corresponds to thelongest element of a Coxeter group.

There are a number of analogies betweenalgebraic groups and Weyl groups – for instance, the number of elements of the symmetric group isn!, and the number of elements of the general linear group over a finite field is related to theq-factorial${\displaystyle [n{]}_q!}$; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by thefield with one element, which considers Weyl groups to be simple algebraic groups over the field with one element.

For a non-abelian connected compact Lie groupG, the firstgroup cohomology of the Weyl groupW with coefficients in the maximal torusT used to define it,^{[note 2]} is related to theouter automorphism group of the normalizer${\displaystyle N=N_G(T),}$ as:^[8]

${\displaystyle \mathrm{Out}(N)\cong H^1(W;T)⋊\mathrm{Out}(G).}$

The outer automorphisms of the group Out(G) are essentially the diagram automorphisms of theDynkin diagram, while the group cohomology is computed inHämmerli, Matthey & Suter 2004 and is a finite elementary abelian 2-group (${\displaystyle (\mathbf{Z}/2{)}^k}$); for simple Lie groups it has order 1, 2, or 4. The 0th and 2nd group cohomology are also closely related to the normalizer.^[8]

• Affine Weyl group
• Semisimple Lie algebra#Cartan subalgebras and root systems
• Maximal torus
• Root system of a semi-simple Lie algebra
• Hasse diagram

• "Coxeter group",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Weisstein, Eric W."Coxeter group".MathWorld.
• Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators

• Finite reflection groups
• Lie algebras
• Lie groups

• Articles with short description
• Short description matches Wikidata