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{{Short description|Subgroup of a root system's isometry group}}
{{Lie groups |Semi-simple}}

In [[mathematics]], in particular the theory of [[Lie algebra]]s, the '''Weyl group''' (named after [[Hermann Weyl]]) of a [[root system]] Φ is a [[subgroup]] of the [[isometry group]] of that root system. Specifically, it is the subgroup which is generated by reflections through the [[hyperplane]]s [[orthogonal]] to at least one of the roots, and as such is a [[finite reflection group]]. In fact it turns out that ''most'' finite reflection groups are Weyl groups.{{sfn|Humphreys|1992|p=6}} Abstractly, Weyl groups are [[finite Coxeter group]]s, and are important examples of these.

The Weyl group of a [[semisimple Lie group]], a semisimple [[Lie algebra]], a semisimple [[linear algebraic group]], etc. is the Weyl group of the [[Root system of a semi-simple Lie algebra|root system of that group or algebra]].

==Definition and examples==
[[File:A2_Weyl_group_(revised).png|thumb|right|The Weyl group of the <math>A_2</math> root system is the symmetry group of an equilateral triangle]]
Let <math>\Phi</math> be a [[root system]] in a Euclidean space <math>V</math>. For each root <math>\alpha\in\Phi</math>, let <math>s_\alpha</math> denote the reflection about the hyperplane perpendicular to <math>\alpha</math>, which is given explicitly as
:<math>s_\alpha(v)=v-2\frac{(v,\alpha)}{(\alpha,\alpha)}\alpha</math>,
where <math>(\cdot,\cdot)</math> is the inner product on <math>V</math>. The Weyl group <math>W</math> of <math>\Phi</math> is the subgroup of the orthogonal group <math>O(V)</math> generated by all the <math>s_\alpha</math>'s. By the definition of a root system, each <math>s_\alpha</math> preserves <math>\Phi</math>, from which it follows that <math>W</math> is a finite group.

In the case of the <math>A_2</math> 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, <math>W</math> 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, <math>W</math> is not the full symmetry group of the root system; a 60-degree rotation preserves <math>\Phi</math> but is not an element of <math>W</math>.

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

==Weyl chambers==
{{see also|Coxeter group#Affine Coxeter groups}}
[[File:Weyl_chambers_for_A2.png|thumb|right|The shaded region is the fundamental Weyl chamber for the base <math>\{\alpha_1,\alpha_2\}</math>]]

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

Since the reflections <math>s_\alpha,\,\alpha\in\Phi</math> preserve <math>\Phi</math>, 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:<ref>{{harvnb|Hall|2015}} Propositions 8.23 and 8.27</ref>
:'''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:<ref>{{harvnb|Hall|2015}} Proposition 8.29</ref>
:'''Theorem''': Fix a Weyl chamber <math>C</math>. Then for all <math>v\in V</math>, the Weyl-orbit of <math>v</math> contains exactly one point in the closure <math>\bar C</math> of <math>C</math>.

==Coxeter group structure==
===Generating set===
A key result about the Weyl group is this:<ref>{{harvnb|Hall|2015}} Propositions 8.24</ref>
:'''Theorem''': If <math>\Delta</math> is base for <math>\Phi</math>, then the Weyl group is generated by the reflections <math>s_\alpha</math> with <math>\alpha</math> in <math>\Delta</math>.
That is to say, the group generated by the reflections <math>s_\alpha,\,\alpha\in\Delta,</math> is the same as the group generated by the reflections <math>s_\alpha,\,\alpha\in\Phi</math>.

===Relations===
Meanwhile, if <math>\alpha</math> and <math>\beta</math> are in <math>\Delta</math>, then the [[Dynkin diagram]] for <math>\Phi</math> relative to the base <math>\Delta</math> tells us something about how the pair <math>\{s_\alpha,s_\beta\}</math> behaves. Specifically, suppose <math>v</math> and <math>v'</math> are the corresponding vertices in the Dynkin diagram. Then we have the following results:
*If there is no bond between <math>v</math> and <math>v'</math>, then <math>s_\alpha</math> and <math>s_\beta</math> commute. Since <math>s_\alpha</math> and <math>s_\beta</math> each have order two, this is equivalent to saying that <math>(s_\alpha s_\beta)^2=1</math>.
*If there is one bond between <math>v</math> and <math>v'</math>, then <math>(s_\alpha s_\beta)^3=1</math>.
*If there are two bonds between <math>v</math> and <math>v'</math>, then <math>(s_\alpha s_\beta)^4=1</math>.
*If there are three bonds between <math>v</math> and <math>v'</math>, then <math>(s_\alpha s_\beta)^6=1</math>.
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 <math>\alpha</math> and <math>\beta</math> are orthogonal, from which it follows easily that the corresponding reflections commute. More generally, the number of bonds determines the angle <math>\theta</math> between the roots. The product of the two reflections is then a rotation by angle <math>2\theta</math> in the plane spanned by <math>\alpha</math> and <math>\beta</math>, as the reader may verify, from which the above claim follows easily.

===As a Coxeter group===
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 group]]s, 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 of a group|presentation]] in which each generator ''x<sub>i</sub>'' is of order two, and the relations other than ''x<sub>i</sub><sup>2</sup>=1'' are of the form (''x''<sub>''i''</sub>''x''<sub>''j''</sub>)<sup>''m''<sub>''ij''</sub></sup>=1.  The generators are the reflections given by simple roots, and ''m<sub>ij</sub>'' 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 <math>W</math> is a Coxeter group, we are saying that those are the ''only'' relations in <math>W</math>.

Weyl groups have a [[Bruhat order]] and [[length function]] in terms of this presentation: the ''[[Length of a Weyl group element|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.

==Weyl groups in algebraic, group-theoretic, and geometric settings==
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. of [[Cartan subalgebra]] for a Lie algebra, of [[maximal torus]] for a Lie group.<ref name="springer">{{Harvnb|Popov|Fedenko|2001}}</ref>

===The Weyl group of a connected compact Lie group===
Let <math>K</math> be a connected compact Lie group and let <math>T</math> be a [[maximal torus]] in <math>K</math>. We then introduce the '''normalizer''' of <math>T</math> in <math>K</math>, denoted <math>N(T)</math> and defined as
:<math>N(T)=\{x\in K|xtx^{-1}\in T,\,\text{for all }t\in T\}</math>.
We also define the '''centralizer''' of <math>T</math> in <math>K</math>, denoted <math>Z(T)</math> and defined as
:<math>Z(T)=\{x\in K|xtx^{-1}=t\,\text{for all }t\in T\}</math>.
The Weyl group <math>W</math> of <math>K</math> (relative to the given maximal torus <math>T</math>) is then defined initially as
:<math>W=N(T)/T</math>.
Eventually, one proves that <math>Z(T)=T</math>,<ref name="Hall 2015">{{harvnb|Hall|2015}} Theorem 11.36</ref> at which point one has an alternative description of the Weyl group as
:<math>W=N(T)/Z(T)</math>.

Now, one can define a root system <math>\Phi</math> associated to the pair <math>(K,T)</math>; the roots are the nonzero [[Weight (representation theory)|weights]] of the adjoint action of <math>T</math> on the Lie algebra of <math>K</math>. For each <math>\alpha\in\Phi</math>, one can construct an element <math>x_\alpha</math> of <math>N(T)</math> whose action on <math>T</math> has the form of reflection.<ref>{{harvnb|Hall|2015}} Propositions 11.35</ref> With a bit more effort, one can show that these reflections generate all of <math>N(T)/Z(T)</math>.<ref name="Hall 2015"/> Thus, in the end, the Weyl group as defined as <math>N(T)/T</math> or <math>N(T)/Z(T)</math> is isomorphic to the Weyl group of the root system <math>\Phi</math>.

===In other settings===

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 [[Semisimple Lie algebra#Cartan subalgebras and root systems|Cartan subalgebra]].

For a [[Lie group]] ''G'' satisfying certain conditions,<ref group="note">Different conditions are sufficient – most simply if ''G'' is connected and either compact, or an affine algebraic group. The definition is simpler for a semisimple (or more generally reductive) Lie group over an [[algebraically closed field]], but a ''relative'' Weyl group can be defined for a [[split Lie group|''split'' Lie group]].</ref> 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<sub>G</sub>''(''T'') by the [[centralizer]] of the torus ''Z'' = ''Z''(''T'') = ''Z<sub>G</sub>''(''T''),

:<math>W(T,G) := N(T)/Z(T).\ </math>

The group ''W'' is finite – ''Z'' is of finite [[Index of a subgroup|index]] in ''N''. If ''T'' = ''T''<sub>0</sub> is a [[maximal torus]] (so it equals its own centralizer: <math>Z(T_0) = T_0</math>) then the resulting quotient ''N''/''Z'' = ''N''/''T'' is called ''the'' [[Maximal torus#Weyl group|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.''<ref name="springer"/>

==Bruhat decomposition==
{{Further|Bruhat decomposition}}

If ''B'' is a [[Borel subgroup]] of ''G'', i.e., a maximal [[connected space|connected]] [[solvable group|solvable]] subgroup and a maximal torus ''T'' = ''T''<sub>0</sub> is chosen to lie in ''B'', then we obtain the [[Bruhat decomposition]]

:<math>G = \bigcup_{w\in W} BwB</math>

which gives rise to the decomposition of the [[flag variety]] ''G''/''B'' into '''Schubert cells''' (see [[Grassmannian]]).

The structure of the [[Hasse 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 by [[Poincaré 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 dimension ''k'' and in dimension ''n'' - ''k'' (where ''n'' 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 the [[longest element of a Coxeter group]].

==Analogy with algebraic groups==
{{Main|q-analog}}
{{See also|Field with one element}}
There are a number of analogies between [[algebraic group]]s and Weyl groups – for instance, the number of elements of the symmetric group is ''n''!, and the number of elements of the general linear group over a finite field is related to the [[q-factorial|''q''-factorial]] <math>[n]_q!</math>; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the [[field with one element]], which considers Weyl groups to be simple algebraic groups over the field with one element.

==Cohomology==
For a non-abelian connected compact Lie group ''G,'' the first [[group cohomology]] of the Weyl group ''W'' with coefficients in the maximal torus ''T'' used to define it,<ref group="note">''W'' acts on ''T'' – that is how it is defined – and the group <math>H^1(W; T)</math> means "with respect to this action".</ref> is related to the [[outer automorphism group]] of the normalizer <math>N = N_G(T),</math> as:<ref name="hms">{{Harvnb|Hämmerli|Matthey|Suter|2004}}</ref>
:<math>\operatorname{Out}(N) \cong H^1(W; T) \rtimes \operatorname{Out}(G).</math>
The outer automorphisms of the group Out(''G'') are essentially the diagram automorphisms of the [[Dynkin diagram]], while the group cohomology is computed in {{Harvnb|Hämmerli|Matthey|Suter|2004}} and is a finite elementary abelian 2-group (<math>(\mathbf{Z}/2)^k</math>); for simple Lie groups it has order 1, 2, or 4. The 0th and 2nd group cohomology are also closely related to the normalizer.<ref name="hms"/>

==See also==
*[[Affine Weyl group]]
*[[Semisimple Lie algebra#Cartan subalgebras and root systems]]
*[[Maximal torus]]
*[[Root system of a semi-simple Lie algebra]]
*[[Hasse diagram]]

==Footnotes==
===Notes===
{{Reflist|group=note}}

===Citations===
{{Reflist|2}}

==References==
{{ref begin}}
{{sfn whitelist|CITEREFPopovFedenko2001}}
*{{Citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction |edition=2nd|series=Graduate Texts in Mathematics|year=2015 |volume=222|publisher=Springer|isbn=978-3-319-13466-6 |url=https://www.springer.com/jp/book/9783319134666}}
*{{Citation | journal = Journal of Lie Theory | volume = 14 | year = 2004 | pages = 583–617 | publisher = Heldermann Verlag | title = Automorphisms of Normalizers of Maximal Tori and First Cohomology of Weyl Groups | first1 = J.-F. | last1 = Hämmerli | first2 = M. | last2 = Matthey | first3 = U. | last3 = Suter | url = http://www.heldermann-verlag.de/jlt/jlt14/mattla2e.pdf |zbl=1092.22004 }}
*{{citation |first=James E. |last=Humphreys |title=Reflection Groups and Coxeter Groups |url=https://books.google.com/books?id=ODfjmOeNLMUC |orig-year=1990 |date=1992 |publisher=Cambridge University Press |isbn=978-0-521-43613-7 |series=Cambridge Studies in Advanced Mathematics |volume=29 |zbl=0725.20028}}
*{{SpringerEOM | title = Weyl group | first1 = V.L. | last1 = Popov | author1-link = Vladimir L. Popov | first2 = A.S. | last2 = Fedenko | year = 2001 }}
{{refend}}

==Further reading==
{{refbegin|2}}
*{{citation |first=Nicolas |last=Bourbaki |title=Lie Groups and Lie Algebras: Chapters 4-6 |url=https://books.google.com/books?id=2CSYFcgAlRMC |year=2002 |publisher=Springer |isbn=978-3-540-42650-9 |series=Elements of Mathematics |zbl=0983.17001}}
*{{citation |first1=Anders |last1=Björner |first2=Francesco |last2=Brenti |title=Combinatorics of Coxeter Groups |url=https://books.google.com/books?id=1TBPz5sd8m0C |year=2005 |publisher=Springer |isbn=978-3-540-27596-1 |series=[[Graduate Texts in Mathematics]] |volume=231 |author1-link=Anders Björner |zbl=1110.05001}}
*{{Citation|author-link=H. S. M. Coxeter|first=H. S. M.|last=Coxeter|title=Discrete groups generated by reflections|journal=Ann. of Math.|volume=35|year=1934|issue=3|pages=588–621|jstor=1968753|doi=10.2307/1968753|citeseerx=10.1.1.128.471}}
*{{Citation|author-link=H. S. M. Coxeter|first=H. S. M.|last=Coxeter|title=The complete enumeration of finite groups of the form <math>r_i^2=(r_ir_j)^{k_{ij}}=1</math>|journal=J. London Math. Soc.|series=1 |volume=10 |issue=1 |doi=10.1112/jlms/s1-10.37.21|year=1935|pages=21–25}}
*{{citation | title = The Geometry and Topology of Coxeter Groups
|first = Michael W. | last = Davis | year = 2007 |publisher = Princeton University Press | url = http://www.math.osu.edu/~mdavis/davisbook.pdf | isbn = 978-0-691-13138-2 |zbl=1142.20020 }}
*{{citation |first2=Clark T. |last2=Benson |first1=Larry C. |last1=Grove |title=Finite Reflection Groups |url=https://books.google.com/books?id=525Gh4uzjnIC |year=1985 |publisher=Springer |series=Graduate texts in mathematics |volume=99 |isbn=978-0-387-96082-1 }}
*{{citation |first=Howard |last=Hiller |title=Geometry of Coxeter groups |url=https://books.google.com/books?id=7jzvAAAAMAAJ |year=1982 |publisher=Pitman |isbn=978-0-273-08517-1 |series=Research Notes in Mathematics |volume=54 |zbl=0483.57002}}
*{{Citation|title=On the Schur Multipliers of Coxeter Groups|first=Robert B.|last=Howlett |journal=J. London Math. Soc.|year=1988|series = 2|volume=38|issue=2|pages=263–276|doi=10.1112/jlms/s2-38.2.263  |zbl=0627.20019}}
*{{Citation|first1 = S.|last1 = Ihara|first2 = Takeo|last2 = Yokonuma|title = On the second cohomology groups (Schur-multipliers) of finite reflection groups|year = 1965|journal = J. Fac. Sci. Univ. Tokyo, Sect. 1|volume = 11|pages = 155–171 |zbl=0136.28802|url = http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6049/1/jfs110203.pdf}}
*{{citation |first=Richard |last=Kane |title=Reflection Groups and Invariant Theory |url=https://books.google.com/books?id=KmL1uuiMyFUC&pg=PP10 |year=2001 |publisher=Springer |series=CMS Books in Mathematics |isbn=978-0-387-98979-2 |zbl=0986.20038 }}
*{{Citation| last=Knapp|first=Anthony W.|title=Lie Groups: Beyond an Introduction|edition=2nd|series=Progress in Mathematics|volume=140|publisher=Birkhaeuser|year=2002|isbn=978-0-8176-4259-4}}
*{{citation| first =E. B. |last=Vinberg| author-link=E. B. Vinberg|title=Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension |journal=Trudy Moskov. Mat. Obshch. |volume=47|year=1984}}
* {{Citation|first = Takeo|last = Yokonuma|title = On the second cohomology groups (Schur-multipliers) of infinite discrete reflection groups|year = 1965|journal = J. Fac. Sci. Univ. Tokyo, Sect. 1|volume = 11|pages = 173–186 |zbl=0136.28803|hdl = 2261/6049}}
{{refend}}

==External links==
* {{springer|title=Coxeter group|id=p/c026980}}
* {{MathWorld | urlname=CoxeterGroup  | title=Coxeter group }}
* {{Citation|url=http://www.jenn3d.org/index.html|title= Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators}}

{{DEFAULTSORT:Weyl Group}}
[[Category:Finite reflection groups]]
[[Category:Lie algebras]]
[[Category:Lie groups]]