All Questions
Tagged with group-theory coxeter-groups
127
questions
1
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0
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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
108
views
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
vote
0
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32
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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
votes
0
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39
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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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0
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59
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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
answer
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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0
answers
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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0
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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
288
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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}) $ ...
- 8,532
1
vote
1
answer
105
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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
$$A\star_{\{(\varphi_1 , C,\varphi_2 )\}}=\langle ...
- 63
0
votes
1
answer
68
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Are complex reflection groups never perfect?
This is a follow-up to
Conceptual reason why Coxeter groups are never simple
A complex reflection group is a finite subgroup of $ U_n $ that is generated by pseudo reflections. A pseudo reflection is ...
- 8,532
4
votes
2
answers
174
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Conceptual reason why Coxeter groups are never simple
Is there a conceptual reason why (non-abelian) Coxeter groups are never simple?
For example is there some obvious normal subgroup that can be defined? Or perhaps it is for some reason clear that the ...
- 8,532
7
votes
1
answer
128
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Condition implying $N(H)/H$ a Coxeter group?
I'm interested in which finite groups can arise as
$$
N(H)/H
$$
for $ H $ a connected subgroup of a compact connected simple Lie group $ G $.
One obvious family of examples is take $ H $ to be the ...
- 8,532
0
votes
1
answer
57
views
For a finite irreducible Coxeter group, what’s the largest set of pairwise-mutually incomparable elements with respect to the weak order?
Given a finite irreducible Coxeter group $W$, what’s the largest subset $K\subseteq W$ such that for all $u,v \in K$, it is not true that
$u <_R v$ (nor $v <_R u$)
where $<_R$ denotes the ...
- 317
1
vote
2
answers
118
views
Number of fixed points of generators of reflections (Coxeter) group
Say I have a group with presentation like
$$\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}...
