All Questions
Tagged with group-theory coxeter-groups
127
questions
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24
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Bruhat Order of the Finite Symmetric Group
I am studying Theorem 2.1.5 in "Combinatorics of Coxeter Groups", but I am confused by a statement in the proof of the "if" direction (the part after $\textbf{However}$). Let me ...
- 63
2
votes
1
answer
109
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Explicit Construction of the Alternating Subgroup of a Coxeter Group
Given a Coxeter group $(W,S)$ with $S=\{s_1,\dots,s_n\}$, I want to show that the alternating subgroup $H$ (containing all elements with even length w.r.t. $S$) is generated by $\{s_is_n\}_{i=1}^{n-1}$...
- 63
1
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0
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37
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Dihedral Groups as Coxeter Groups
I have just completed Exercise 1.2 in the book "Combinatorics of Coxeter Groups" stated below:
Show that there exist Coxeter systems $(W,S)$ and $(W',S')$ with $|S|\neq|S'|$ such that $W\...
- 63
2
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0
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41
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Relationship between the Strong and Weak Exchange Property of Coxeter Groups
I am a beginner at studying Coxeter group theory, and I am confused with the strong and weak exchange property when reading the book "Combinatorics of Coxeter Groups". Let me first state the ...
- 63
1
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61
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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 ...
- 115
0
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1
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15
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Suzuki Coxeter groups proof queries
I am reading a proof over 4.2 in Suzuki group theory I and can't make sense of some parts.
I will just type the proof and then say what my query is.
Statement
Let $(W,S)$ be a Coxeter system. Let $T$ ...
- 1,828
0
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20
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Coxeter groups - Suzuki group theory I
At the start of the section Coxeter groups in Suzuki's "Group theory I", we have Coxeter group W with generating set $S$. we have $T$ as the set of elements of $W$ that are conjugate to some ...
- 1,828
7
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108
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Proving that this relation implies another relation on the Coxeter group [4,3,3,4].
I have a group with five generators $\sigma_i$, and the following relations:
\begin{split}
\sigma_i^2 = \varepsilon \\
|i-j| \neq 1 \implies (\sigma_i\sigma_j)^2 = \varepsilon \\
(\sigma_0\sigma_1)^4 =...
- 2,258
4
votes
2
answers
296
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Signed Permutations and Coxeter Groups
Context: (most of which is pulled from comments and answers to https://mathoverflow.net/questions/431964/signed-permutations-and-operatornameson)
The diagonal subgroup $ C_2^n $ of $ O_n(\mathbb{Z}) $ ...
- 8,542
1
vote
1
answer
106
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Why isn't a coxeter group a HNN-extension?
A doctorate told me to think about why there is no mapping from coxeter groups to $\mathbb{Z}$. This makes sense since HNN-extensions are of the form
$$A\star_{\{(\varphi_1 , C,\varphi_2 )\}}=\langle ...
- 63
0
votes
1
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70
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Are complex reflection groups never perfect?
This is a follow-up to
Conceptual reason why Coxeter groups are never simple
A complex reflection group is a finite subgroup of $ U_n $ that is generated by pseudo reflections. A pseudo reflection is ...
- 8,542
4
votes
2
answers
175
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Conceptual reason why Coxeter groups are never simple
Is there a conceptual reason why (non-abelian) Coxeter groups are never simple?
For example is there some obvious normal subgroup that can be defined? Or perhaps it is for some reason clear that the ...
- 8,542
7
votes
1
answer
128
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Condition implying $N(H)/H$ a Coxeter group?
I'm interested in which finite groups can arise as
$$
N(H)/H
$$
for $ H $ a connected subgroup of a compact connected simple Lie group $ G $.
One obvious family of examples is take $ H $ to be the ...
- 8,542
0
votes
1
answer
57
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For a finite irreducible Coxeter group, what’s the largest set of pairwise-mutually incomparable elements with respect to the weak order?
Given a finite irreducible Coxeter group $W$, what’s the largest subset $K\subseteq W$ such that for all $u,v \in K$, it is not true that
$u <_R v$ (nor $v <_R u$)
where $<_R$ denotes the ...
- 317
1
vote
2
answers
119
views
Number of fixed points of generators of reflections (Coxeter) group
Say I have a group with presentation like
$$\langle s,t,u \mid s^2,t^2,u^2,(st)^2,(su)^3,(ut)^4\rangle,$$
faithful on set $S$ with exactly one orbit ($|S|$ is known). How could I determine $|\text{Fix}...
0
votes
0
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58
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Coxeter Groups Classification Proof
I have recently noticed that Bourbaki's proof (Chapter 6, section 4, no.1, Lemma 10, on p205) of the classification of finite Coxeter groups uses the inequality
$$ \frac{1}{1+p} + \frac{1}{1+q} + \...
- 168
4
votes
0
answers
65
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Is there an algorithm to check that a subgroup of a CAT$(0)$ group is *not* quasiconvex?
Let $G$ be a finitely generated CAT$(0)$ group and $H$ a subgroup. If $H$ is quasiconvex then it is finitely generated, so we can immediately conclude that any non-finitely generated subgroup of $G$ ...
- 2,382
3
votes
1
answer
72
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Davis Regular Tessellations of Spheres and Straight Line Coxeter Groups
In Davis' "Geometry and Topology of Coxeter Groups", section B.3, in particular Theorem B.3.1, there is a proof that every finite "straight line" Coxeter group is associated to a ...
- 168
0
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0
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55
views
Why a retraction from a building to an apartment is not isometric?
Let $X$ be an affine building and $\mathcal{A}$ a system of apartment. For any apartment $A\in \mathcal{A}$ and a chamber $C$ in $A$, let us consider the retraction $\rho=\rho_{A,C}\colon X\...
- 161
2
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0
answers
63
views
When does multiplying by an involution increase the Bruhat Order in the Symmetric Group?
Let $w \in \mathrm{Sym}(n)$ for some positive integer $n$. Let $r$ be an involution in $\mathrm{Sym}(n)$, and write it as the product of disjoint transpositions like so: $$r = \prod_{i=1}^k (a_i,b_i) $...
- 317
1
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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
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1
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175
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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
2
votes
1
answer
64
views
Does the Coxeter group $C(D_n)$ have any "proper reflection quotients" except $C(A_{n-1})$?
Here, a reflection quotient is a surjective homomorphism between Coxeter groups mapping reflections to reflections. A reflection quotient $C(\Delta) \to C(\Gamma)$ is proper if it is not injective and ...
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1
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1
answer
290
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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
0
answers
100
views
Classification of Finite Coxeter Groups Bjorner
While Humphreys gives a classification of finite irreducible Coxeter groups by their "geometric representation", Bjorner and Brenti (Combinatorics of Coxeter Groups) leave it as an exercise ...
- 168
1
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1
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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
2
votes
1
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133
views
How to find the longest element in a double coset of a Weyl group using SageMath?
I asked a question about computing longest element in a double coset in AskSage
It has not been answered for a long time. So I asked it here.
Let $W$ be a finite Coxeter group. Denote by $W_I$ the ...
- 14.6k
1
vote
1
answer
232
views
Defining the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in GAP
I am looking to define the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in the software GAP. In general, one can define these groups separately. For example, let's say
...
- 4,125
2
votes
0
answers
22
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System of representatives in reflection groups and subgroups
I'm working on a paper from Steinberg (1974), "On a theorem of Pittie". The paper is mainly about roots and reflection groups. I'm having trouble understanding the proof of lemma 2.5(a) ...
- 21
2
votes
2
answers
53
views
Generating sets of semi-direct products with $\mathbb{Z}_2$
Suppose a group $G$ splits as a semidirect product $N\rtimes\mathbb{Z}_2$, and let $\phi:G\to\mathbb{Z}_2$ the the associated quotient map. If I have a subset of elements $\{g_1,\dots,g_n,h\}$ of $G$ ...
- 2,382