All Questions
23
questions
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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
4
votes
2
answers
296
views
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
0
votes
1
answer
70
views
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
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 ...
- 4,786
1
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0
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141
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Which Coxeter Elements have powers that are the longest element of the (Finite, Irreducible) Coxeter Group?
Let $(W,S)$ be a finite, irreducible Coxeter Group. I thought it was true (from Humphrey's book Ex 2 on page 82) that if the Coxeter Number of $W$, $h$, is even then $$c^{h/2} = \omega_0 \quad \dagger$...
- 317
3
votes
1
answer
57
views
Partial Order on a Vector space with a Crystallographic Coxeter Group
Let $V$ be a vector space, and $W$ an irreducible, crystallographic Coxeter group on $V$ with simple root system $\Pi = \{a_1 \cdots a_n \}$.
We define a partial ordering on $V$ by $u \geq v$ iff $u - ...
- 127
2
votes
1
answer
274
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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
1
vote
0
answers
70
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Counting inversions of random elements in coxeter groups
I am trying to find a general interperetation to the following facts (pls be patient to read it).
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of ...
2
votes
1
answer
644
views
Coxeter length in the symmetric group equals number of inversions
Let $S_n$ be the symmetric group on the set $\{1,\dots, n\}$ and $\sigma\in S_n$. An inversion of $\sigma$ is a pair $(i,j)$ such that $1\leq i<j\leq n$ and $\sigma(i)>\sigma(j)$. Let $S_n$ be ...
- 6,323
1
vote
1
answer
104
views
Orbits in Coxeter Group
I'm interested in computing the orbits in a finite coxeter group, i.e. $D_4$ (see here
).
The orbits of the roots $\mathcal{O}(r_i)$ can be obtained by applying all the reflections
$\displaystyle ...
- 760
1
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0
answers
56
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Finite group with $BN$-pair, if $s_i$ and $s_j$ are conjugate in $W$, why do they have the same index parameter?
Suppose $G$ is a finite group with $BN$-pair, and $(W,S)$ is its Coxeter system.
Iwahori's theorem that the corresponding Hecke algebra $\mathcal{H}$ has a standard basis $\{a_w:w\in W\}$ where $a_w=...
- 1,673
4
votes
1
answer
754
views
The number of reduced expressions for the longest element of $B_n$?
Let $W=W_{\Phi}$ be a reflection group, with root system $\Phi$, and $\Delta=\{\alpha_1, ...,\alpha_n\}\subseteq \Phi$ a simple system. So $W$ is generated by the $s_{\alpha_i}=s_i$ for $i=1,2,...n. $ ...
- 984
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