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 $\{s_is_n\}_{i=1}^{n-1}$...
- 135k
1
vote
0
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36
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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
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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
4
votes
2
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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
3
votes
0
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193
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No cycles in finite coxeter graphs
Is there an elementary (no consideration of root systems involved) proof of the fact that the graph of an finite coxeter system doesn't entail any cycle? I got as far as this: If there were any cycle $...
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
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
votes
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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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
1
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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
$$A\star_{\{(\varphi_1 , C,\varphi_2 )\}}=\langle ...
- 31.3k
0
votes
1
answer
70
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Are complex reflection groups never perfect?
This is a follow-up to
Conceptual reason why Coxeter groups are never simple
A complex reflection group is a finite subgroup of $ U_n $ that is generated by pseudo reflections. A pseudo reflection is ...
- 45.7k
7
votes
1
answer
128
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Condition implying $N(H)/H$ a Coxeter group?
I'm interested in which finite groups can arise as
$$
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 ...
- 11.2k
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 ...
- 9,798
0
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1
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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 ...
- 973