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...

set. Separately, the symmetric group belongs to a larger family of reflection groups called complex reflection groups, which are defined in terms of their...

the reflection hyperplanes pass through the origin). The corresponding notions can be defined over other fields, leading to complex reflection groups and...

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,...

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...

Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However...

symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their...

vectors, twice 63). The thirty-sixth-largest of thirty-seven total complex reflection groups is W ( E 7 ) {\displaystyle W(E_{7})} , with order 2 63 {\displaystyle...

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...

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...

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...

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...

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...

in invariant theory of finite groups, began the study of complex polytopes, and classified the complex reflection groups. Shephard earned his Ph.D. in...

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...

(with Robert Howlett), and the determination of the action of a complex reflection group on the cohomology of the complement of its reflecting hyperplanes...

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...

Commensurability (group theory) Compact group Compactly generated group Complete group Complex reflection group Congruence subgroup Continuous symmetry...

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...

mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest...

\mathbb {Z} } -reflection groups. Simple exotic p-compact groups are again in 1-1-correspondence with irreducible complex reflection groups whose character...

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...

translations, rotations, reflections, and glide reflections (see below § Classification). The set of Euclidean plane isometries forms a group under composition:...

indexes makes 73 the only Sheldon prime. There are precisely 37 complex reflection groups. In three-dimensional space, the most uniform solids are: the...

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...

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...

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...

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...

as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation)...

definitions exist only for the regular complex polyhedra, whose symmetry groups are complex reflection groups. The complex polyhedra are mathematically more...