All Questions
Tagged with group-theory coxeter-groups
127
questions
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About parabolic subgroup of a Weyl group
Let $W$ be a Weyl group/Coxeter group. Let $\Phi$ be the associated root system, fix a positive root system $\Phi^+$ and let
$\Delta$ be the set of simple roots.
Let $W_I$ be the parabolic subgroup ...
- 255
2
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0
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117
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Reflection length of the longest word in a Coxeter group.
Let $(W,S)$ be a Coxeter system. The set of reflections of $W$ is $S=\{wsw^{-1}:s \in S, w \in W\}$. The reflection length of $w \in W$ is defined as the minimal number $m$ such that $w = r_1 \cdots ...
- 14.6k
2
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1
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646
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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
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315
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Why $(1+x)(1+x+x^2)\cdots(1+x+x^2+\cdots+x^n)$ gives the Poincare series?
I am looking for an explanation of the fact that the polynomial $$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$ with Mahonian numbers gives the Poincare series for the symmetric group $S_{n+1}$ considered as a ...
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$B$-adapted pairs and $\hat{G}$ acting on $W$
Let $(G,B,N,S)$ be a Tits system, and let $\varphi: G \rightarrow \hat{G}$ be a $B$-adapted homomorphism. This means the kernel of $\varphi$ is contained in $G$, and for every $g \in \hat{G}$, there ...
- 34.3k
7
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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
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131
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In what different terms can Coxeter systems be described?
My starting point is this question: https://mathoverflow.net/questions/214569
As I understand it they say, that the Coxeter matrix is not sufficient to describe the group.
I thought that up to ...
- 103
4
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1
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121
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Cardinality of a coxeter group
Let ${G}$ be a Coxeter group with the next presentation
\begin{equation}
G = \left\langle s_1,s_2,\cdots,s_{n-1} : (s_is_{i+1})^3=1 , \ (s_is_j)^2=1 \ ,\ |i-j| > 1 \right\rangle
\end{equation}
...
4
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1
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359
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Conditions for a neat subgroup to act fixed-point free
Given a hyperbolic reflection group $G$ acting on hyperbolic space $\mathbb{H}_n$ by, well, reflections in hyperplanes.
Does a neat subgroup of $G$ act fixed-point free on $\mathbb{H}_n$? If not, ...
user149890
3
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169
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Longest element of a subgroup
Say I have a finite Weyl group, $W$, and a set of generators $S:= \{s_1,...,s_k\}$ (making $W,S$ a coxeter system) and an automorphism $\theta: W\rightarrow W$ which permutes $S$. I know that the ...
- 5,993
4
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1
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216
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Question about proof of positive roots under reflection
Let $(W, S)$ be a finite Coxeter system. Furthermore, let $V$ be a real vector space with a (finite) basis $\{ \alpha_s | s \in S \}$.
For every $s \in S$ one can define the reflection $\sigma_s : V ...
- 3,149
3
votes
1
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230
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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
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How to justify that any set of coxeter generators for a Weyl group are simple
Let $G$ be a group with a $\mathrm{BN}$-pair, and let $W:=N/(B\cap N)$ be the Weyl group of $G$, where $W$ is generated by a set $S$ of simple roots (as in the definition of a $\mathrm{BN}$-pair) and $...
- 387
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Alcoves in extended affine Weyl group of $A_1^{(1,1)}$
For affine Coxeter/Weyl groups, there is a standard representation into $GL_n(\mathbb{R})$ (where $n$ is the rank of the Coxeter group) where the reflecting hyperplanes form a nice hyperplane ...
- 11
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104
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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