Search results

...thor2=Smith, Larry |name-list-style=amp | title=Invariant Theory of Finite Groups | location=Providence, RI | publisher=American Mathematical Society | year= ...ar transformation]] <math>g: V\to V</math> such that the order of ''g'' is finite and the '''fixed subspace''' <math>V^g = \{ v\in V:\ gv=v \} </math> of all ...

...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 ...

...ance, <math>B</math> is positive definite if and only if <math>W</math> is finite. It is a [[faithful representation]] of <math>W</math>. 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 ...

...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 == ...

==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 ...

...c group by Nagata's theorem.) The finite generation is easily seen for a [[finite group]] ''G'' acting on a [[finitely generated algebra]] ''R'': since ''R'' ...on needed|date=September 2015}}<!--Let ''G'' be a finite group and ''V'' a finite-dimensional ''G''-module. Then ''G'' acts on the [[ring of polynomial funct ...

...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 ...

...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: ...

...gram''' ('''SLP''') for a finite group ''G'' =&nbsp;&lang;''S''&rang; is a finite sequence ''L'' of elements of ''G'' such that every element of ''L'' either ...matrix group properties. Babai and Szemerédi prove that every element of a finite group ''G'' has an SLP of length ''O''(log²|''G''|) in every gen ...

...njectured the following extension of Mehta's integral to all [[Root system|finite root systems]], Mehta's original case corresponding to the ''A''_{''n''& ...'} are the degrees of the generators of the ring of invariants of the reflection group. ...

...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 ...

...p scheme]]s.<ref>[[Eric Friedlander]] and [[Andrei Suslin]], Cohomology of finite group schemes over a field. ''Inventiones Mathematicae'' 127 (1997), 209--2 ...gical algebra|homological]] properties. In particular, Schur algebras have finite [[global dimension]]. ...

...ort description|Theorem describing fusion of elements in Sylow subgroup of finite group}} ...rious non-[[simple group|simplicity]] criteria focussing on showing that a finite group has a [[normal subgroup]] of [[Index of a subgroup|index]] ''p''. ...

...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. ...

...] and [[Lie algebra]]s. For the topics in the representation theory of Lie groups and Lie algebras, see [[Glossary of representation theory]]. Because of the {{Lie groups}}'''Notations''': ...

...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 ...

While 1-polytopes can have unlimited ''p'', finite regular complex polygons, excluding the double prism polygons <sub>''p''</s ====12 Irreducible Shephard groups==== ...

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