Questions tagged [reflection-group]
Apt for questions related to reflection group (discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space).
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The rotation symmetry group and the reflection group: Is there a name for what they have in common?
When I turn on my monitor, the brand name fills the screen. But since I mounted my monitor upside down so I can watch it in bed looking up, the power-on screen is upside down.
But I noticed that it ...
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Why is pointwise stabilizer of a reflecting hyperplane a cyclic group?
Let $G$ be a pseudoreflection group and $H$ be a reflecting hyperplane. Let $G_H$ be the subgroup of $G$ consisting of all those elements of $G$ which stabilize the reflecting hyperplane $H$ pointwise....
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Finite cyclic groups are reflection groups.
I am following a note on complex reflections. There the notion of pseudoreflection groups are introduced. A pseuforeflection $\sigma : \mathbb C^n \longrightarrow \mathbb C^n$ is a linear isomorphism ...
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Restriction of a reflection group to a reflection hyperplane
Let $G$ be a finite reflection group acting on a euclidean vector space $V$ of dimension $n$. Choose any parabolic subgroup $G'$ and let $H_{G'}$ be the subspace of points fixed under $G'$ and $\...
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what is the exact relation between Dedekind tessellation and modular group
In the wiki for the Modular Group the group ($\Gamma$) is described as a $(2,3,\infty)$.
The page has a diagram for "a typical fundamental domain for the action of $\Gamma$ on the upper half-...
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In a finite reflection group, an involution is a product of commuting reflections
I am working through the book Reflection Groups and Coxeter Groups by Humphreys. I got stuck while trying Exercise 1.12.3:
If $w \in W$ is an involution, prove that $w$ can be written as a product of ...
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Young Diagrams and Branching Symmetric irreps to Dihedral irreps
In this question I asked whether there exists some formula that computes the multiplicity of the irreps occuring in the branching rules from $S_n$ to $C_n = \langle r : r^n = 1 \rangle$ where we embed ...
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Testing if an element is in the derived subgroup
Let $ G $ be a finite group. Let $ G':=[G,G] $ be the derived subgroup. How do you test if an element $ g \in G $ is in $ G' $?
I'm specifically interested in how to do this in GAP. In other words I ...
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Are complex reflection groups never perfect?
This is a follow-up to
Conceptual reason why Coxeter groups are never simple
A complex reflection group is a finite subgroup of $ U_n $ that is generated by pseudo reflections. A pseudo reflection is ...
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Smallest group that is not a complex reflection group
What is the smallest group which is not a complex reflection group?
Many well known families of finite groups are complex reflection groups https://en.wikipedia.org/wiki/Complex_reflection_group For ...
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How do roots of a root system correspond to symmetric points of the associated Weyl group
So I've read that the Weyl group of the $A_3$ root system is the full symmetry group of a tetrahedron $T_h$, $C_3$ is $O_h$, and $H_3$ is $I_h$. I've read some complicated proofs of this but I don't ...
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Generators of the monodromy generating braid groups
I'm reading the article 'Complex reflection groups, braid groups and Hecke algebras', by Broué, Malle and Rouquier, but I need some help with the 'generators of the monodromy' they defined and that ...
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how to find reduced words for each element of reflection (Coxeter) group in GAP
I have a finite reflection (or Coxeter) group defined abstractly through the standard presentation
$$(s_i s_j)^{c_{ij}}=1$$
For each of its elements I want to find the number of reduced words equal to ...
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120 cell generated from quaternions
The two quaternions
$\omega={1\over 2}(-1,1,1,1)$ and $q={1\over 4}(0,2,\sqrt{5}+1,\sqrt{5}-1)$ generate a finite group under multiplication with 120 elements that form the vertices of a 600 cell, ...
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Angles of the Fundamental Alcove (Chamber?)
I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I ...
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