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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
Commensurability (group theory) Compact group Compactly generated group Complete group Complex reflection group Congruence subgroup Continuous symmetry
List_of_group_theory_topics
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
Number with a real and an imaginary part
the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis. The complex numbers form a rich structure
Complex_number
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
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
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
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)
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
pseudoreflection generalizes the concepts of reflection and complex reflection and is simply called reflection by some mathematicians. It plays an important
Pseudoreflection
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)
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 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
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
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
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
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
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
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
South Korean girl group
Music Awards to selects 'Trend of the Year' for the first time...100% reflection of fan votes [Official]] (in Korean). Sports Seoul. Archived from the
Illit
Concept in military and political science
The expression military–industrial complex (MIC) describes the relationship between a country's military and the defense industry that supplies it, seen
Military–industrial_complex
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
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
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
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
Material which alters light reflection or transmission on optics
that falls on them. More complex optical coatings exhibit high reflection over some range of wavelengths, and anti-reflection over another range, allowing
Optical_coating
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
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
Real square matrix whose columns and rows are orthogonal unit vectors
bring any orthogonal matrix to the identity; thus an orthogonal group is a reflection group. The last column can be fixed to any unit vector, and each choice
Orthogonal_matrix
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
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
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
Molecular cloud complex in the constellation Monoceros
Solar System. The most notable feature of the complex is the presence of a large number of reflection nebulae, illuminated by the hot, blue stars of
Monoceros_R2
to Ariki-Koike algebras (which are q-deformations of certain complex reflection groups). The study of these various classes of generalizations forms
Schur_algebra
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
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
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
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
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)
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
Mathematical structure
simplicial complex A corresponds to the standard geometric realization of W. Standard generators of the Coxeter group are given by the reflections in the
Building_(mathematics)
Function that is its own inverse
negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry;
Involution_(mathematics)
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
Italian historian (1939–2026)
a new trial. His book was not only about Sofri but was also a general reflection on the scientific methods used by a historian, and their similarity to
Carlo_Ginzburg
1656 painting by Diego Velázquez
viewer, although some scholars have speculated that their image is a reflection from the painting Velázquez is shown working on. The Italian Baroque painter
Las_Meninas
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
Group homomorphism into the general linear group over a vector space
relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups allow many group-theoretic problems to
Group_representation
American streamer collective
with some feeling it showed a deeper reflection of power dynamics and a lack of empathy. In July 2025, the group went live for 12 hours each day that
AMP_(streamer_collective)
Attribution of the U.S.'s high incarceration rate to profit
from the original on March 30, 2018. "Masked Racism: Reflections on the Prison Industrial Complex". History Is A Weapon. Archived from the original on
Prison–industrial_complex
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
Girl/Female
Tamil
Reflection, Image, Radiance
Boy/Male
Tamil
Gunjik | கà¯à®¨à¯à®œà¯€à®•
Reflection
Gunjik | கà¯à®¨à¯à®œà¯€à®•
Boy/Male
Hindu, Indian
Perception; Reflection
Boy/Male
Hindu, Indian
Our Reflection
Girl/Female
Hindu, Indian
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’.
Girl/Female
Japanese
Mirror reflection.
Girl/Female
Bengali, Hindu, Indian
Reflection; Mirror
Girl/Female
Indian, Malayalam
Reflection
Boy/Male
Buddhist, Indian, Japanese
Ancient Reflection
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Tamil, Telugu
Reflection; Outlook; Reflection 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’.
Boy/Male
Hindu
Reflection
Boy/Male
Tamil
Reflection
Boy/Male
Indian
Reflection; Gnawing Reflection
Boy/Male
Hindu
Reflection
Boy/Male
Hindu, Indian, Punjabi, Sanskrit, Sikh
Thought; Reflection
Boy/Male
Bengali, Hindu, Indian
Image; Reflection
Girl/Female
Arabic, Assamese, Australian, Hindu, Indian, Marathi, Muslim, Sindhi
Mirror; Reflection
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
Girl/Female
Latin
Mother of Aeacus.
Boy/Male
Arabic, Muslim, Sindhi
Narrator of Hadith
Girl/Female
Arabic, Muslim
A Young Dog or Fox; First Umayyad Khalifah
Boy/Male
Arabic, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Telugu
Ocean
Boy/Male
Hindu
Lotus
Girl/Female
Hindu
Conceived in the mind
Boy/Male
Hindu, Indian
Briliant
Surname or Lastname
English
English : probably a variant of Sudbury.
Girl/Female
Hindu, Indian
Intelligent; Part of Bhim; Good
Girl/Female
Norse
Fate.
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
COMPLEX REFLECTION-GROUP
a.
Not complex; uncompounded; simple.
a.
Intricate; entangled; complicated; complex.
n.
Election beforehand.
a.
The act of choosing; choice; selection.
n.
An image given back from a reflecting surface; a reflected counterpart.
n.
A part reflected, or turned back, at an angle; as, the reflection of a membrane.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
n.
The act of reflecting, or turning or sending back, or the state of being reflected.
n.
Want of reflection.
a.
Throwing back images; as, a reflective mirror.
n.
Election a second time, or anew; as, the reelection of a former chief.
adv.
In a complex manner; not simply.
a.
Complex, complicated.
n.
A reflecting telescope.
n.
That which is produced by reflection.
a.
Given to reflection or serious consideration; reflective; contemplative; as, a reflecting mind.
a.
Repeatedly compound; made up of complex constituents.
n.
A device for reflecting sound.
n.
The return of rays, beams, sound, or the like, from a surface. See Angle of reflection, below.
n.
See Reflection.