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Concept in mathematics
mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial
Complex_reflection_group
Discrete group type in group theory
the reflection hyperplanes pass through the origin). The corresponding notions can be defined over other fields, leading to complex reflection groups and
Reflection_group
Mathematical group
reflection group. Every real reflection group can be complexified to give a complex reflection group, so the complex reflection groups form another generalization
Parabolic subgroup of a reflection group
Parabolic_subgroup_of_a_reflection_group
Group that admits a formal description in terms of reflections
Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However
Coxeter_group
Number line and triangular tiling's symmetry mathematical structure
symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their
Affine_symmetric_group
Topics referred to by the same term
mathematics, complex group may refer to: An archaic name for the symplectic group Complex reflection group A complex algebraic group A complex Lie group This
Complex_group
Subgroup of a root system's isometry group
finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple
Weyl_group
In mathematics, Mitchell's group is a complex reflection group in 6 complex dimensions of order 108 × 9!, introduced by Mitchell (1914). It has the structure
Mitchell's_group
Pictorial representation of symmetry
complex polygon is not called a Coxeter group, but instead a Shephard group, a type of Complex reflection group. The order of p[q]r is 8 / q ⋅ ( 1 / p
Coxeter–Dynkin_diagram
four lines through each point. Its complex reflection group is 3[3]3[3]3 or , order 648, also called a Hessian group. It has 27 copies of , order 24, at
Hessian_polyhedron
Group of symmetries of a regular polygon
mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest
Dihedral_group
Nonabelian group in algebraic group theory
by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3
Binary_tetrahedral_group
Australian mathematician
(with Robert Howlett), and the determination of the action of a complex reflection group on the cohomology of the complement of its reflecting hyperplanes
Gustav_Lehrer
German mathematician
question of whether every finite complex reflection group is a Weyl group of an object analogous to a finite group of Lie type. They baptized the unknown
Gunter_Malle
Concept in algebraic topology
\mathbb {Z} } -reflection groups. Simple exotic p-compact groups are again in 1-1-correspondence with irreducible complex reflection groups whose character
P-compact_group
group is a complex reflection group, 3[3]3[3]3 or of order 648, and the product of this with a group of order 2 is another complex reflection group,
Hessian_group
Commensurability (group theory) Compact group Compactly generated group Complete group Complex reflection group Congruence subgroup Continuous symmetry
List_of_group_theory_topics
Group of symmetries of an n-dimensional hypercube
r>2} , these groups are no longer real reflection groups; instead, they belong to the infinite family of imprimitive complex reflection groups. In particular
Hyperoctahedral_group
Natural number
indexes makes 73 the only Sheldon prime. There are precisely 37 complex reflection groups. In three-dimensional space, the most uniform solids are: the
37_(number)
British mathematician (1927–2016)
in invariant theory of finite groups, began the study of complex polytopes, and classified the complex reflection groups. Shephard earned his Ph.D. in
Geoffrey_Colin_Shephard
a group of order 2 is a 3-dimensional complex reflection group of order 2160 generated by 45 complex reflections of order 2. The invariants form a polynomial
Valentiner_group
Generalization of a polytope in real space
symmetry. For any regular polytope the symmetry group (here a complex reflection group, called a Shephard group) acts transitively on the flags, that is, on
Complex_polytope
Isometry of the Eluclidean plane
translations, rotations, reflections, and glide reflections (see below § Classification). The set of Euclidean plane isometries forms a group under composition:
Euclidean_plane_isometry
case when K is the field C of complex numbers, the first condition is usually stated as "G is a complex reflection group". Shephard and Todd derived a
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Non-commutative group with 6 elements
one with a 3-fold rotation axis in a plane of reflection (and hence also in two other planes of reflection): C3v Consider three colored blocks (red, green
Dihedral_group_of_order_6
Type of group in mathematics
product of SO(n) and any subgroup formed with the identity and a reflection. The group with two elements {±I} (where I is the identity matrix) is a normal
Orthogonal_group
Type of symmetry group
and reflection in the horizontal axis (isomorphic to C2, the cyclic group of order 2). the groups each consisting of the identity and reflection in a
Frieze_group
finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was
Rank_3_permutation_group
construct analogues of Lie groups over finite fields, called Chevalley groups. complex reflection group complex reflection group coroot coroot Coxeter 1
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Emission nebula in the constellation Orion
or Sharpless 279) is an HII region and bright nebulae that includes a reflection nebula located in the constellation Orion. It is the northernmost part
Sh_2-279
ix}{\begin{matrix}g/r&r\\p&g/p\end{matrix}}\end{bmatrix}}} The complex reflection group is p[q]r, order g = 8 / q ⋅ ( 1 / p + 2 / q + 1 / r − 1 ) − 2 {\displaystyle
Configuration_(polytope)
Dutch mathematician
(Heckman-Opdam hypergeometric functions), and with Dunkl operators on complex reflection groups. Opdam was an invited speaker in 2000 with talk Hecke algebras
Eric_M._Opdam
Set with associative invertible operation
two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed
Group_(mathematics)
Simplicial complex
Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the
Coxeter_complex
automorphism group of this complex lattice has index 2 in the full automorphism group of the Coxeter–Todd lattice and is a complex reflection group (number
Coxeter–Todd_lattice
groups, or fields, and many other things. List of mathematical examples List of algebraic surfaces List of curves List of complex reflection groups List
Lists_of_mathematics_topics
Uniform 6-polytope
Hess. It has 27 vertices, 72 3-edges, and 27 3{3}3 faces. Its complex reflection group is 3[3]3[3]3, order 648. The 221 is fourth in a dimensional series
2_21_polytope
Dutch mathematician
Utrecht University who worked on linear algebraic groups, Hecke algebras, complex reflection groups, and who introduced Springer representations and the
T._A._Springer
Flat-sided three-dimensional shape
definitions exist only for the regular complex polyhedra, whose symmetry groups are complex reflection groups. The complex polyhedra are mathematically more
Polyhedron
British geometer
lattice. In 1954 he and G. C. Shephard classified the finite complex reflection groups. In March 1948 he was elected a Fellow of the Royal Society. Scholia
J._A._Todd
Inflated feelings of personal ability, privilege, or infallibility
A god complex is an unshakable belief characterized by consistently inflated feelings of personal ability, privilege, or infallibility. The person is
God_complex
Reflection nebula in the constellation of Orion
as M78 or NGC 2068) is a reflection nebula in the constellation Orion. It is the brightest diffuse reflection nebula in a group that includes NGC 2064,
Messier_78
Star-forming region in the constellation Orion
processes involved in stellar formation, though the complex contains dark nebulae, emission nebulae, reflection nebulae, and H II regions. The presence of ripples
Orion_molecular_cloud_complex
Classification system for symmetry groups in geometry
is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing
Coxeter_notation
Uniform 6-polytope
space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is 3[3]3[4]2, order 1296. It has a half-symmetry quasiregular construction
1_22_polytope
Four-dimensional analog of the icosahedron
4-dimensional space. Both have 120 vertices, and 120 edges. The first has Complex reflection group 3[5]3, order 360, and the second has symmetry 5[3]5, order 600
600-cell
Mathematical abelian group
as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation)
Klein_four-group
Capacity of humans to exercise introspection
Self-reflection is the ability to witness and evaluate one's own cognitive, emotional, and behavioural processes. In psychology, other terms used for this
Self-reflection
2015 studio album by Fifth Harmony
Reflection is the debut studio album by American girl group Fifth Harmony. It was released on January 30, 2015, by Syco Music and Epic Records. Lyrically
Reflection (Fifth Harmony album)
Reflection_(Fifth_Harmony_album)
Excessive preoccupation with oneself
the Greek mythological figure Narcissus who fell in love with his own reflection, narcissism has evolved into a psychological concept studied extensively
Narcissism
Shape with four equal sides and angles
ISBN 978-1-938664-45-8. Grove, L. C.; Benson, C. T. (1985). Finite Reflection Groups. Graduate Texts in Mathematics. Vol. 99 (2nd ed.). New York: Springer-Verlag
Square
Group of transformations under which the object is invariant
detail. The isometry groups in one dimension are: the trivial cyclic group C1 the groups of two elements generated by a reflection; they are isomorphic
Symmetry_group
Type of group in abstract algebra
the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an
Symmetric_group
American mathematician (born 1941)
Opdam: Dunkl, C. F.; Opdam, E. M. (2003). "Dunkl operators for complex reflection groups". Proc. London Math. Soc. 86: 70–108. arXiv:math/0108185. doi:10
Charles_F._Dunkl
Electrical engineers graphical calculator
to the reference impedance, Z0. The Smith chart is plotted on the complex reflection coefficient plane in two dimensions and may be scaled in normalised
Smith_chart
Casino resort in Winchester, Nevada, US
Winchester, Nevada, United States. It is owned and operated by Genting Group as part of the Resorts World brand. It had been the site of the Stardust
Resorts_World_Las_Vegas
Groups of point isometries in 3 dimensions
reflections, equation 12.61 Burban, Igor. "Du Val Singularities" (PDF). Coxeter, H. S. M. (1974), "7 The Binary Polyhedral Groups", Regular Complex Polytopes
Point groups in three dimensions
Point_groups_in_three_dimensions
Undergraduate mathematics textbook
of reflection systems and reflection groups, the special case of dihedral groups, and root systems. Part III of the book concerns Coxeter complexes, and
Mirrors_and_Reflections
Reflection nebula in the constellation Orion
NGC 2071 is a reflection nebula in the constellation Orion. It was discovered on January 1, 1784, by William Herschel. It is part of a group of nebulae that
NGC_2071
Reflection nebula in the constellation Orion
NGC 2064 (also designated as LBN 1627) is a reflection nebula that is located 1600 light years away from earth in the constellation Orion. It was discovered
NGC_2064
Jungian psychological concept
In neo-Freudian psychology, the Electra complex, as proposed by Swiss psychiatrist and psychoanalyst Carl Jung in his Theory of Psychoanalysis, is a girl's
Electra_complex
American girl group
2015 – Complex". Complex. Archived from the original on June 23, 2015. Retrieved August 9, 2015. Cox, Jamieson. "Review: Fifth Harmony's Reflection Has Many
Fifth_Harmony
Branch of mathematics that studies the properties of groups
finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects
Group_theory
Core pattern of emotions, memories, perceptions, and desires
A complex is a structure in the unconscious that is objectified as an underlying theme—like a power or a status—by grouping clusters of emotions, memories
Complex_(psychology)
Application of Clifford algebra
reflections as basic elements, and constructs all other transformations and geometric objects out of them. Formally: it identifies planar reflections
Plane-based_geometric_algebra
Real square matrix whose columns and rows are orthogonal unit vectors
therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation. The
Orthogonal_matrix
Russian mathematician
Russian Academy of Sciences. Popov, Vladimir L. (1982). Discrete complex reflection groups. Utrecht: Communications of the Mathematical Institute Rijksuniversiteit
Vladimir Popov (mathematician)
Vladimir_Popov_(mathematician)
Geometric arrangements of points, foundational to Lie theory
of Φ. As it acts faithfully on the finite set Φ, the Weyl group is always finite. The reflection planes are the hyperplanes perpendicular to the roots, indicated
Root_system
South Korean girl group
girl groups have released music together. Illit debuted with a concept based on the "authentic feelings of teenage girls", differing from "complex" backstories
Illit
pseudoreflection generalizes the concepts of reflection and complex reflection and is simply called reflection by some mathematicians. It plays an important
Pseudoreflection
Peripheral nebular regions of the Orion Molecular Cloud Complex
X-ray sources. NGC 1788 is a small reflection nebula located about 6° west of the densest regions of the Orion complex, on the border between the constellations
Peripheral nebular regions of the Orion Complex
Peripheral_nebular_regions_of_the_Orion_Complex
Subgroup of the Clifford algebra associated to a quadratic space
preimage of a reflection squares to ±1 ∈ Ker (Spin(V) → SO(V)), and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in
Pin_group
Exploration of subsurface properties with seismology
Reflection seismology (or seismic reflection) is a method of exploration geophysics that uses the principles of seismology to estimate the properties of
Reflection_seismology
Geometrical property
operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance
Symmetry_(geometry)
to Ariki-Koike algebras (which are q-deformations of certain complex reflection groups). The study of these various classes of generalizations forms
Schur_algebra
Type of number in combinatorial mathematics and statistics
Gordon, Ian G.; Griffeth, Stephen (2012). "Catalan numbers for complex reflection groups". American Journal of Mathematics. 134 (6): 1491–1502. arXiv:0912
Fuss–Catalan_number
Universal construction of a complex Lie group from a real Lie group
the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification
Complexification_(Lie_group)
Property in optics
reaching the interface, as well as the critical angle for total internal reflection, their intensity (Fresnel equations) and Brewster's angle. The refractive
Refractive_index
Revision of the C++ programming language released in 2026
"Language Evolution" working group of the C++ Standards Committee, described the potential impacts for including reflection to C++ as a "whole new language
C++26
Concept in military and political science
Military–Industrial Complex. Harper & Rowe. pp. 280–285. ISBN 978-0-06-045296-4. SBN 06-045296-X. Kennan, George Frost (1997). At a Century's Ending: Reflections 1982–1995
Military–industrial_complex
Group of geologic formations in New Mexico, US
The Vadito Group is a group of geologic formations that crops out in most of the Precambrian-cored uplifts of northern New Mexico. Detrital zircon geochronology
Vadito_Group
Reflection nebula in the constellation Orion
NGC 2067 is a reflection nebula in the constellation Orion. It was discovered in 1876 by Wilhelm Tempel. It is part of a group of nebulae that also includes
NGC_2067
Isometric automorphisms of a hyperbolic space
freedom. Orientation reversing reflection through a line — one reflection; two degrees of freedom. combined reflection through a line and translation
Hyperbolic_motion
2026 studio album by Drake
becoming a bit of a problem. His output is mirroring his lack of self-reflection". Fitzgerald concluded her review noting that "Iceman is Drake's unearned
Iceman_(album)
Extension of the factorial function
function to complex numbers. First studied by Daniel Bernoulli, the gamma function Γ ( z ) {\displaystyle \Gamma (z)} is defined for all complex numbers z
Gamma_function
2015–16 concert tour by Fifth Harmony
The Reflection Tour was the fourth concert tour by American girl group Fifth Harmony. Visiting Europe, North America and Asia (one date in Adu Dhabi),
The_Reflection_Tour
Technique in artificial intelligence
minimize errors (like hallucinations) and increase interpretability. This reflection is a form of "test-time compute", where additional computational resources
Feedback_neural_network
Concept in linear algebra
(also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing
Householder_transformation
Rational function of the form (az + b)/(cz + d)
transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2, C). Together with
Möbius_transformation
2001 studio album by Tool
ninth track incorrectly spelled as "Lateralis". The original title of "Reflection" was "Resolution" before being changed three months prior to the album's
Lateralus
Group of rotations in 3 dimensions
orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a
3D_rotation_group
Circle with radius of one
{\textstyle x^{2}+y^{2}=1.} Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the
Unit_circle
Canadian experimental rock duo
the chest"), and was initially suggested as a joke; this later became a reflection of the band's sound, characterized by the members as "dissonance-induced
Angine_de_Poitrine
Human blood group system involving 49 blood antigens
terminology) is not an accurate reflection of the antigens encountered since many (e.g. Rh38) have been combined, reassigned to other groups, or otherwise removed
Rh_blood_group_system
hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which
Hurwitz_surface
Mathematical group that can be generated as the set of powers of a single element
representation theory of more general finite groups. In the complex case, a representation of a cyclic group decomposes into a direct sum of linear characters
Cyclic_group
Capacity for introspection and individuation as a subject
In the philosophy of self, self-awareness is the awareness and reflection of one's own personality or individuality, including traits, feelings, and behaviors
Self-awareness
Mathematical group
2000.8494, MR 1805481. Simons, Christopher Smyth (1997), Hyperbolic reflection groups, completely replicable functions, the Monster and the Bimonster, Ph
Bimonster_group
Theorem in algebraic geometry
conformal automorphisms (in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and of dimension one for a torus)
Hurwitz's automorphisms theorem
Hurwitz's_automorphisms_theorem
United Kingdom and have created complex social frameworks. The definition of ethnicity has been defined as "the social group a person belongs to, and either
Ethnic groups in the United Kingdom
Ethnic_groups_in_the_United_Kingdom
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
Boy/Male
Hindu, Indian, Punjabi, Sanskrit, Sikh
Thought; Reflection
Girl/Female
Hindu, Indian
Reflection
Boy/Male
Buddhist, Indian, Japanese
Ancient Reflection
Boy/Male
Hindu, Indian
Perception; Reflection
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
Boy/Male
Tamil
Gunjik | கà¯à®¨à¯à®œà¯€à®•
Reflection
Gunjik | கà¯à®¨à¯à®œà¯€à®•
Boy/Male
Hindu, Indian
Our Reflection
Boy/Male
Hindu
Reflection
Boy/Male
Bengali, Hindu, Indian
Image; Reflection
Girl/Female
Japanese
Mirror reflection.
Boy/Male
Indian
Reflection; Gnawing Reflection
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Tamil, Telugu
Reflection; Outlook; Reflection Reflection
Boy/Male
Tamil
Reflection
Girl/Female
Arabic, Assamese, Australian, Hindu, Indian, Marathi, Muslim, Sindhi
Mirror; Reflection
Girl/Female
Indian, Malayalam
Reflection
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from any of various places called Copley, for example in County Durham, Staffordshire, and Yorkshire, from the Old English personal name Coppa (apparently a byname for a tall man) or from copp ‘hilltop’ + lēah ‘woodland clearing’.
Girl/Female
Tamil
Reflection, Image, Radiance
Boy/Male
Hindu
Reflection
Girl/Female
Bengali, Hindu, Indian
Reflection; Mirror
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
Boy/Male
Gujarati, Hindu, Indian, Kannada
Brother of Prahlad
Girl/Female
American, Armenian, Australian, British, Chinese, Christian, Danish, Dutch, English, Finnish, French, German, Greek, Italian, Jamaican, Swedish, Swiss
A Crown of Garland; Crown; Garland; Crowned in Victory; Victorious
Girl/Female
Indian
Boy/Male
Gujarati, Indian
See
Boy/Male
Tamil
Gaze
Girl/Female
Indian
The greatness, Blessing of Allah
Surname or Lastname
English
English : variant of Schofield.
Girl/Female
Hebrew
Green.
Girl/Female
Tamil
Rajdulari | ராஜதà¯à®²à®¾à®°à¯€
Dear princess
Female
Native American
Native American Sioux name HANTAYWEE means "faithful."
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
a.
The act of choosing; choice; selection.
n.
See Reflection.
n.
A reflecting telescope.
a.
Repeatedly compound; made up of complex constituents.
a.
Given to reflection or serious consideration; reflective; contemplative; as, a reflecting mind.
n.
Election a second time, or anew; as, the reelection of a former chief.
n.
An image given back from a reflecting surface; a reflected counterpart.
a.
Intricate; entangled; complicated; complex.
a.
Throwing back images; as, a reflective mirror.
n.
That which is produced by reflection.
n.
Want of reflection.
n.
The return of rays, beams, sound, or the like, from a surface. See Angle of reflection, below.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
n.
A part reflected, or turned back, at an angle; as, the reflection of a membrane.
adv.
In a complex manner; not simply.
n.
The act of reflecting, or turning or sending back, or the state of being reflected.
n.
Election beforehand.
n.
A device for reflecting sound.
a.
Complex, complicated.
a.
Not complex; uncompounded; simple.