# Questions tagged [coxeter-groups]

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

265 questions
Filter by
Sorted by
Tagged with
32 views

### 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 ...
23 views

### Does support of an inversion root of an X-reduced word always intersects complement of X?

Suppose that we have a Coxeter system $(W,S)$ with an associated reduced root system $R$ and let $X$ be a subset of $S$. Is it true that for any $(\varnothing,X)$-reduced word $w \in W$ support of an ...
15 views

### Exploring Polygon Drawing in Hyperbolic Space with Integer Factor Angles: Seeking Simpler or Standard Methods

I am currently attempting to draw polygons in hyperbolic space with angles that are integer multiples of pi, i.e., $\pi / k$ for k an integer. I am particularly interested in determining the vertex ...
• 726
40 views

119 views

### Explicit expression for simple roots of the root systems $A_2$, $B_2$ and $G_2$ in 2D

I often find explicit expression for rank-2 root systems as $A_2$, $B_2$ and $G_2$ in a 3D Euclidean space. Does anybody have an explicit expression for the simple roots in terms of $e_1,e_2$ in the ...
• 9,244
1 vote
37 views

### Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model

According to P. Francesco et al. conformal field theory book the conformal charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which ...
1 vote
57 views

### 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 ...
95 views

• 801
67 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 ...
• 8,514
167 views

### 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 ...
• 8,514
128 views

### 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 ...
• 8,514
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 ...
82 views

### How do roots of a root system correspond to symmetric points of the associated Weyl group

So I've read that the Weyl group of the $A_3$ root system is the full symmetry group of a tetrahedron $T_h$, $C_3$ is $O_h$, and $H_3$ is $I_h$. I've read some complicated proofs of this but I don't ...
1 vote
115 views

124 views

### 120 cell generated from quaternions

The two quaternions $\omega={1\over 2}(-1,1,1,1)$ and $q={1\over 4}(0,2,\sqrt{5}+1,\sqrt{5}-1)$ generate a finite group under multiplication with 120 elements that form the vertices of a 600 cell, ...
• 218
87 views

### Number of inversions in $S_n$

I have a problem about some result appearing just before the proposition 1.5.2 in Combinatorics of Coxeter Groups by Bjorner, Brenti. It's about the inversions in $S_n$. I don't understand why 1.26) ...
• 35
1 vote
64 views

### Angles of the Fundamental Alcove (Chamber?)

I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I ...
• 8,020
1 vote
166 views

### Reflection Group of Type $D_n$

Here is the description of the reflection group of type $D_n$ in Humphreys' book Reflection Groups and Coxeter Groups: ($D_n$, $n \ge 4)$ Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of ...
• 8,020
63 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 vote
276 views

### Reflection Group of Type $C_n$

In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way: Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of ...
• 8,020
1 vote
93 views

### Classification of Finite Coxeter Groups Bjorner

While Humphreys gives a classification of finite irreducible Coxeter groups by their "geometric representation", Bjorner and Brenti (Combinatorics of Coxeter Groups) leave it as an exercise ...
• 168
50 views

### Generating function for cardinality of Bruhat intervals

For a finite Coxeter system $(W, S)$, the Poincare polynomial counts the number of elements of a certain length: $$P_W(t) = \sum_{x \in W} t^{l(x)}.$$ The polynomials for types $A$, $B$, and $D$ are ...
• 13.1k
1 vote
54 views

### Affine Permutations as a Coxeter Group

I was stuck trying to show that the group of $n$-affine permutations (i.e. bijections of $\mathbb{Z}$ such that $\sum_{i=1}^{n} f(i)-i =0$ and $f(i+n)=f(i)+n$ for a certain $n \in \mathbb{N}$) is a ...
• 403
1 vote
161 views

### Simple Reflections on Simple Roots

I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a ...
• 8,020
Given a subgroup $G$ of $O(V)$ that is generated by reflections ($V$ a Euclidean space), under what conditions can we determine that $G$ is finite? I realize that if the angles between any two roots ...