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...thor2=Smith, Larry |name-list-style=amp | title=Invariant Theory of Finite Groups | location=Providence, RI | publisher=American Mathematical Society | year= A diagonalizable pseudoreflection is sometimes called a '''semisimple reflection'''. ...

...b, \ldots\}</math> and <math>\times</math> denotes the [[direct product of groups]]. Abstractly, <math>S_\lambda</math> is isomorphic to the product <math>S ...\ldots, (n - 1 \ n)</math>.<ref>{{citation |title=Combinatorics of Coxeter groups |last1=Björner |first1=Anders |author1-link=Anders Björner |last2=Brenti |f ...

...risovich Vinberg]], for finding a [[fundamental domain]] of a [[hyperbolic reflection group]]. Let <math>\Gamma < \mathrm{Isom}(\mathbb{H}^n)</math> be a hyperbolic reflection group. Choose any point <math>v_0 \in \mathbb{H}^n</math>; we shall call it ...

...trefoils stick number.png|175px|thumb|Square knot {{=}} trefoil + trefoil reflection. [[stick number|Sticks depicted]].]] ...|connected sum]] of a [[trefoil knot]] with its [[Reflection (mathematics)|reflection]]. It is closely related to the [[granny knot (mathematics)|granny knot]], ...

...epresentation has several foundational properties in the theory of Coxeter groups; for instance, <math>B</math> is positive definite if and only if <math>W</ This representation describes <math>W</math> as a [[reflection group]], with the caveat that <math>B</math> might not be positive definite ...

...46 |url=https://www.cambridge.org/core/books/reflection-groups-and-coxeter-groups/2910C1E00877D33A04A512791B6EDD72|last1=Humphreys |first1=James E. |isbn=978 *Arithmetic Groups, [[Lecture Notes in Mathematics]] 789, Springer Verlag 1980 (from lectures ...

==Reflection group== ...rs) having inner product –1 with ''w''=(0,0,1) are the simple roots of the reflection group. These are the vectors (λ,1,λ²/2–1) for λ in the Leech lat ...

...bolic subgroups essentially coincide in the case of finite real reflection groups. Parabolic subgroups arise in the theory of [[algebraic group]]s, through == Background: reflection groups == ...

| thesis_title = Smooth Actions of the Classical Groups ...7–451|via=Project Euclid|doi=10.1215/S0012-7094-91-06217-4}}</ref> and the reflection group trick. ...

...lfs |first1=Martin |last2=De Keninck |first2=Steven |title=Graded Symmetry Groups: Plane and Simple |url=https://www.researchgate.net/publication/353116859 } ...composition because these factors are the invariants of the <math>k</math>-reflection <math>R \in \text{Pin}(p,q,r)</math>. A well known special case is the [[Ch ...

...roup (mathematics)|groups]] that may be sub-groups of other transformation groups. ...between opposite group actions because [[commutative group]]s are the only groups for which these opposites are equal. ...

...is abelian group is a [[Klein four-group]]-module, where the group acts by reflection in each of the coordinate directions (here depicted by red and blue arrows ...module''' consists of<ref>{{Citation|title=Representation Theory of Finite Groups and Associative Algebras|year=1962|last1=Curtis|last2=Reiner|first1=Charles ...

...dels''' provide a model of the earth's response to [[seismic wave]]s. In [[reflection seismology]], the '''anelastic attenuation factor''', often expressed as '' ...umber]] and i is the imaginary unit. Reflection [[seismogram]]s record the reflection wave along the propagation path r from the source to reflector and back to ...

...group has order 48 as shown by these [[Möbius triangles|spherical triangle reflection domains]]. ...]]. It was named by [[Alfred Young (mathematician)|Alfred Young]] in 1930. Groups of this type are identified by a parameter {{mvar|n}}, the dimension of the ...

[[File:Glide reflection.svg|right|thumb|300px|A [[glide reflection]] is a type of Euclidean motion.]] ...er motions are motions like [[reflection (geometry)|reflection]]s, [[glide reflection]]s and [[Improper rotation]]s that invert the [[orientation (mathematics)|o ...

...osite.<ref>Atoji, A. (1965). ''Graphical representations of magnetic space groups'', American Journal of Physics, '''33'''(3), 212–219, {{doi|10.1119/1.19713 ...nstruct an anti-mirror reflection (m') from a [[Reflection symmetry|mirror reflection]] (m) and an anti-identity operation (1') executed in either order. ...

...al of the [[sporadic simple group]]s were discovered as rank 3 permutation groups. The primitive rank 3 permutation groups are all in one of the following classes: ...

==Compact Lie groups== ...on a generalization of the [[Jacobi eigenvalue algorithm]] to compact Lie groups. ...

...'</sub> are the degrees of the generators of the ring of invariants of the reflection group. ...vtxt|Opdam|1989}} gave a uniform proof for all crystallographic reflection groups.<ref>{{cite journal|last=Opdam|first=E.M.|year=1989|title=Some applications ...

...of polynomials]] over the ring of polynomials invariant under the [[finite reflection group]] of a [[root system]]. If the reflection group ''W'' corresponds to the [[Weyl group]] of a compact [[semisimple gro ...