Weyl group

GPTKB entity

Statements (388)
Predicate Object
gptkbp:instanceOf gptkb:algebraic_geometry
gptkb:group_of_people
gptkb:mathematical_concept
graph
mathematical diagram
gptkbp:acts_by orthogonal transformations
linear transformations
gptkbp:actsOn gptkb:Vector
gptkb:root
permutations
reflections
gptkbp:alternativeName Cartan_matrix
Cayley_graph
Chevalley_group
Coxeter_diagram
Coxeter_group
Dynkin_diagram
Galois_group
Mathieu_group
group_of_Lie_type
parabolic_subgroup
reflection_group
root_lattice
von_Dyck_group
gptkbp:application gptkb:Lie_theory
gptkb:algebraic_geometry
gptkb:geometry
gptkb:hyperbolic_geometry
gptkb:Lie_algebras
crystallography
physics
combinatorics
expander graphs
random walks
classification of regular polytopes
network design
interconnection networks
symmetry of tilings
gptkbp:appliesTo field extensions
gptkbp:associatedWith gptkb:Weyl_group
gptkbp:builtBy gptkb:Chevalley_basis
gptkbp:category field theory
group theory
algebraic structures
presentation of a group
discrete group
gptkbp:class gptkb:affine_reflection_group
gptkb:complex_reflection_group
gptkb:real_reflection_group
classified by Dynkin diagrams
finite reflection group
finite root systems
finite-dimensional semisimple Lie algebras
simple Lie algebras over complex numbers
untwisted and twisted types
gptkbp:containsElement reflection
gptkbp:definedIn generators and relations
arbitrary field
group generated by reflections
gptkbp:defines lattice generated by the roots of a root system
A von Dyck group is a group with presentation D(p, q, r) = ⟨ x, y | x^p = y^q = (xy)^r = 1 ⟩ for integers p, q, r ≥ 2.
gptkbp:describedBy gptkb:Weyl_group
gptkb:Coxeter_matrix
gptkbp:describes gptkb:root
structure of a group
field automorphisms
generators and relations of Coxeter group
gptkbp:determinant is nonzero
gptkbp:diagonal_entries are 2
gptkbp:discoveredBy gptkb:Émile_Mathieu
gptkbp:discoveredIn 1861
gptkbp:edge multiplication by generator
relation between generators
gptkbp:edgeLabelRepresents order of product of generators
gptkbp:encodes root system information
gptkbp:entries_are integers
gptkbp:example gptkb:Suzuki_group
gptkb:Klein_four-group
gptkb:Weyl_group
gptkb:group_of_people
gptkb:symmetric_group_S_n
gptkb:E6(q)
gptkb:E7(q)
gptkb:E8(q)
gptkb:F4(q)
gptkb:PΩ(n,q)
gptkb:PSL(n,q)
gptkb:PSU(n,q)
gptkb:PSp(2n,q)
gptkb:hypercube_graph
gptkb:finite_Coxeter_group
gptkb:A_n_root_lattice
gptkb:D_n_root_lattice
gptkb:E_8_root_lattice
gptkb:Ree_group
graph
dihedral group
hyperoctahedral group
cyclic group
cycle graph
dihedral group graph
G2(q)
D(2,3,4) is the symmetry group of the square
D(2,3,5) is the symmetry group of the regular icosahedron
D(2,3,3) is the symmetry group of the equilateral triangle
gptkbp:field gptkb:Lie_theory
gptkb:algebra
gptkb:geometry
gptkb:mathematics
gptkb:Galois_theory
abstract algebra
group theory
gptkbp:fieldOfStudy gptkb:algebra
group theory
finite group theory
gptkbp:generalizes gptkb:Schreier_graph
orthogonal group
gptkbp:hasApplication gptkb:algebraic_geometry
gptkb:geometry
gptkb:theoretical_physics
gptkb:topology
gptkb:Chevalley_groups
gptkb:Schubert_calculus
coding theory
cryptography
crystallography
modular forms
number theory
particle physics
physics
representation theory
combinatorics
automorphic forms
flag varieties
kac–moody algebras
finite simple group classification
gptkbp:hasEdge angle between roots
multiplicity of root lengths
gptkbp:hasEdgeLabel integer greater than or equal to 3
gptkbp:hasExceptionalType gptkb:E_6
gptkb:E_7
gptkb:E_8
gptkb:G_2
F_4
gptkbp:hasInvariant diameter
automorphism group
girth
spectrum
gptkbp:hasMember gptkb:M11
gptkb:M23
gptkb:M12
gptkb:M22
gptkb:M24
gptkbp:hasNode simple root
generator of group
gptkbp:hasProperty gptkb:symmetry
orthogonal group
can be crystallographic or non-crystallographic
can be finite or infinite
can be infinite
can be reducible or irreducible
generated by involutions
presentation by relations
many are simple groups
finite for finite extensions
many are simple except for small parameters
exceptional types correspond to exceptional Lie algebras
arise from automorphisms of algebraic groups
can be constructed via generators and relations
includes both classical and exceptional groups
most are non-abelian
order depends on field and type
twisted types involve field or graph automorphisms
classical types correspond to classical Lie algebras
constructed from algebraic groups over finite fields
gptkbp:hasSpecialCase gptkb:Weyl_group
regular graph
vertex-transitive graph
gptkbp:hasSubfield gptkb:Lie_theory
gptkb:geometry
group theory
lattice theory
gptkbp:hasSubgroup gptkb:Weyl_group
gptkb:group_of_people
gptkb:maximal_torus
dihedral group
Borel subgroup
gptkbp:hasType gptkb:C_n
gptkb:E_6
gptkb:E_7
gptkb:E_8
gptkb:G_2
gptkb:affine_Dynkin_diagram
gptkb:extended_Dynkin_diagram
gptkb:twisted_affine_Dynkin_diagram
A_n
B_n
D_n
F_4
classical group
exceptional group
non-simply laced
simply laced
gptkbp:heldBy finite reflection group
generalized in Kac–Moody algebras
invertible for semisimple Lie algebras
not necessarily symmetric
square matrix
symmetrizable
used in classification of simple Lie algebras
used in the theory of quantum groups
used to define Serre relations
https://www.w3.org/2000/01/rdf-schema#label Weyl group
gptkbp:importantFor classification of finite simple groups
gptkbp:includes gptkb:Suzuki_group
gptkb:Weyl_group
gptkb:group_of_people
gptkb:affine_Coxeter_group
gptkb:crystallographic_Coxeter_group
gptkb:finite_Coxeter_group
gptkb:infinite_Coxeter_group
gptkb:Steinberg_group
gptkb:Ree_group
gptkb:twisted_Chevalley_group
dihedral group
hyperoctahedral group
gptkbp:introducedIn 1878
1930s
1934
1947
1955
gptkbp:isA gptkb:Lie_group
simple group
gptkbp:isDirected can be directed or undirected
gptkbp:isFinite if defined over finite field
if defined over infinite field
gptkbp:isGraph yes
gptkbp:isOneOf26SporadicGroups true
gptkbp:isSimple yes
gptkbp:isUndirected yes
gptkbp:loveInterest braid relations
order-2 relations
gptkbp:namedAfter gptkb:Évariste_Galois
gptkb:Claude_Chevalley
gptkb:Hermann_Weyl
gptkb:Sophus_Lie
gptkb:Walther_von_Dyck
gptkb:Arthur_Cayley
gptkb:Émile_Mathieu
gptkb:Élie_Cartan
gptkb:H._S._M._Coxeter
gptkb:Eugene_Dynkin
gptkbp:nodeRepresents generator
gptkbp:notableFor first discovered sporadic simple groups
gptkbp:notablePerson gptkb:Sophus_Lie
gptkb:Wilhelm_Killing
gptkb:Élie_Cartan
gptkb:Ludwig_Schläfli
gptkb:H.S.M._Coxeter
gptkbp:notation gptkb:Gal(E/F)
[p,q,...]
D(p, q, r)
gptkbp:numberOfTeams 5
gptkbp:off-diagonal_entries are non-positive integers
gptkbp:order finite or infinite
finite
depends on type and field
gptkbp:orderOfM11 7920
gptkbp:orderOfM12 95040
gptkbp:orderOfM22 443520
gptkbp:orderOfM23 10200960
gptkbp:orderOfM24 244823040
gptkbp:originatedIn gptkb:semisimple_Lie_group
semisimple Lie algebra
algebraic group over finite field
gptkbp:partOf sporadic simple group
gptkbp:partOfSeries gptkb:C_n
A_n
B_n
D_n
gptkbp:property regular graph
can be finite or infinite
generated by involutions
vertex-transitive
connected if generators generate the group
even lattice (for simply-laced types)
integral lattice
unimodular (for E_8)
can be classified
preserves inner product
finite if 1/p + 1/q + 1/r > 1
infinite if 1/p + 1/q + 1/r ≤ 1
gptkbp:publishedIn Eugene Dynkin's papers
gptkbp:relatedConcept gptkb:Weyl_group
gptkb:Euclidean_group
gptkb:symmetry
gptkb:Brouwer_fixed-point_theorem
gptkb:mirror_symmetry
gptkb:crystallographic_group
Penrose tiling
hyperplane
orthogonal group
fundamental domain
Platonic solid
regular polytope
gptkbp:relatedTo gptkb:Schreier_graph
gptkb:Trinity
gptkb:Weyl_group
gptkb:algebraic_geometry
gptkb:group_of_people
gptkb:symmetry
gptkb:Killing_form
gptkb:Lie_group
gptkb:root
gptkb:ADE_classification
gptkb:Artin_group
gptkb:Coxeter_element
gptkb:Coxeter_number
gptkb:fundamental_theorem_of_Galois_theory
gptkb:Chevalley_basis
gptkb:Coxeter–Dynkin_diagram
gptkb:McKay_correspondence
Cartan subalgebra
modular group
finite field
automorphism group
quiver representation
weight lattice
normal extension
permutation group
separable extension
group action
gptkbp:represents gptkb:Weyl_group
gptkbp:seeAlso modular group
gptkbp:structure gptkb:group_of_people
gptkbp:studied_since 19th century
gptkbp:studiedBy gptkb:mathematician
gptkb:Wilhelm_Killing
gptkb:Élie_Cartan
gptkb:Jacques_Tits
gptkb:H._S._M._Coxeter
gptkb:Ludwig_Schläfli
gptkbp:studiedIn gptkb:algebraic_graph_theory
geometric group theory
group theory
algebraic combinatorics
combinatorial group theory
crystal bases
modern algebra
quantum groups
gptkbp:subclassOf gptkb:group_of_people
gptkb:Lie_group
gptkbp:used_in gptkb:Kac–Moody_algebras
representation theory
Lie algebra theory
gptkbp:used_to_classify gptkb:semisimple_Lie_algebras
gptkbp:usedFor solvability by radicals
gptkbp:usedIn gptkb:Weyl_group
gptkb:algebra
gptkb:algebraic_geometry
gptkb:geometry
gptkb:theoretical_physics
gptkb:Lie_group
gptkb:Kac–Moody_algebra
crystallography
group theory
representation theory
theoretical computer science
combinatorics
algebraic groups
semisimple Lie algebra
algebraic group theory
singularity theory
classification of finite simple groups
classification of regular polytopes
Lie algebra classification
algebraic group classification
solvability of polynomials
classification of tessellations
gptkbp:usedToClassify gptkb:finite_reflection_groups
field extensions
regular polytopes
gptkbp:vertexRepresents group element
gptkbp:visualizes labeled graph
gptkbp:bfsParent gptkb:Hermann_Weyl
gptkb:Trinity
gptkb:group_of_people
gptkbp:bfsLayer 4