All Questions
20
questions
1
vote
0
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65
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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 ...
- 8,040
1
vote
1
answer
175
views
Reflection Group of Type $D_n$
Here is the description of the reflection group of type $D_n$ in Humphreys' book Reflection Groups and Coxeter Groups:
($D_n$, $n \ge 4)$ Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of ...
- 8,040
1
vote
1
answer
290
views
Reflection Group of Type $C_n$
In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way:
Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of ...
- 8,040
1
vote
1
answer
168
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Simple Reflections on Simple Roots
I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a ...
- 8,040
4
votes
2
answers
115
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Does every reflection generating set of a RA Coxeter group contain a conjugate of every standard generator?
I am interested in understanding generating sets of right-angle Coxeter groups (RACGs) consisting of reflections. More precisely, let $(W,S)$ be a finite rank RACG, and write $R=\{wsw^{-1}\mid s\in S\;...
- 2,382
4
votes
0
answers
233
views
How to find generators of translation subgroup of an abstract reflection (coxeter) group
I have an infinite reflection group https://en.wikipedia.org/wiki/Coxeter_group
Take for example the affine groups $[4,4],[4,3,4],[4,3,3,4]$...
I'd like to get an explicit expression for generators ...
- 1,010
0
votes
0
answers
63
views
Coxeter graph of the group $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$
I am reading the first chapter of Combinatorics of Coxeter Groups by A.Björner and F.Brenti. In the first example they say that the graph with $n$ isolated vertices (no edges) is the Coxeter graph of ...
- 621
2
votes
1
answer
274
views
Motivation for definition of bilinear form in linear representation of Coxeter groups?
In a set of notes on Coxeter groups I am reading the following definitions are made:
Let $M = (m_{ij})_{1 \leq i,j \leq n}$ be a symmetric $n \times n$ matrix with entries from $\mathbb{N} \cup \...
- 2,501
3
votes
0
answers
79
views
How would one classify point groups?
By point group I mean a finit subgroup of $\mathrm O(\Bbb R^n)$.
Lists of point groups for some small dimensions are found on Wikipedia, but I am not certain about their completeness. As there seem ...
- 30.1k
7
votes
1
answer
2k
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What is a Coxeter Group?
I've recently started investigating abstract algebra and have now stumbled upon "Coxeter Groups", which are a mystery to me.
I've read that Coxeter Groups
have something to do with reflections (in ...
- 559
3
votes
1
answer
230
views
What does it mean for a Coxeter system to be of "spherical" type?
In the theorem of the paper Sur les valeurs propres de la transformation de Coxeter the author uses in the main theorem the term "spherical" to refer to a property that Coxeter systems $(W,S)$ can ...
- 959
1
vote
1
answer
44
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Confused by reflection formula for Coxeter systems of type $I_2(4)$ and $I_2(6)$ in Humphreys.
In Section 5.3 of Humphreys book on Coxeter/Reflection groups, he develops a geometric representation of a Coxeter system $(W,S)$ by taking an $\mathbb{R}$-vector spaces $V$ with basis $\{\alpha_s:s\...
- 997
-1
votes
2
answers
792
views
How many reflection subgroups are in $D_{2n}$?
Given the dihedral group $D_{2n}$ of order $2n$, is there a formula for the number of reflection subgroups of $D_{2n}$?
- 571
1
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0
answers
64
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Fundamental Region $F$ for Coxeter Group $G\subseteq\mathcal{O}(\mathbb{R}^3)$
Let $G\subseteq \mathcal{O}(\mathbb{R}^3)$ (orthogonal transformations). For a reflection $S\in G$ through a hyperplane $\mathcal{P}\subset\mathbb{R}^3$ we call the two unit vectors $\pm r$ that are ...
- 760
3
votes
1
answer
1k
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Finding the longest element $w_0$ of the reflection group $S_n$ with respect to the positive system $\{e_i - e_j, 1≤i < j≤1\}$
Preliminaries
Let $V = \mathbb R^n$, and $W = S_n$ the symmetric group. We consider the set $\Phi = \{±(e_i - e_j) \mid 1 ≤ i < j ≤ n\} $ (where the $e_i$ are the standard vectors), which is a ...
- 3,057