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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
|