# Questions tagged [coxeter-groups]

For questions about Coxeter groups, an abstract group that admits a formal description in terms of reflections.

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### How to translate an element in a Coxeter group written as a matrix in Sage to reflections (a list)?

I am trying to use Sage to reduce a word to a reduced word. For example, consider the word $w=[4, 3, 2, 4, 3, 2, 1, 2, 4, 3, 2, 1, 3]=s_4s_3s_2s_4s_3s_2s_1s_2s_4s_3s_2s_1$. I used the following code ...
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### Fundamental domains in Reflection groups and Coxeter groups - by Humphreys

In this thm I do not understand how he is using the induction in item d). The step t=1 is clear. enter image description here enter image description here
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### How to obtain uniqueness in correspondence between simple systems and positive systems?

In reading the appendix of Lectures on Chevalley Groups by Steinberg, I'm having trouble understanding the uniqueness aspect of Proposition 9 (in both parts). Here is the setup. Let $V$ be an inner ...
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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 ...
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### The coefficient of a root in a root system must be 0 or at least 1

Let W be a Coxeter group (not necessarily finite), and let Π and Φ be the corresponding root basis and root system. Suppose that x ∈ Φ + and a ∈ Π such that the coefficient of a in x is not zero. ...
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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} ...
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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, ...
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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 ...
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### Is this “co-location” of two $E_6$'s, two $F_4$'s, and one $E_8$ possible?

Edited 1/5/2018 4pm US EDST: At the bottom of this post, I have added an email to Dr. Klizting - I sent this email to him after he posted his answer to the question. Question: Imagine three sets of ...
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### Reduced decomposition of the long element of $C_n$ or $B_n$?

Let $W$ be the Weyl group of the root system of type $C_n$. Then $W$ can be identified with the group of signed permutations on $1, 2, ... , n$. Let $S = \{s_1, ... , s_n\}$, where $s_i$ swaps $i$ ...
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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 ...
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