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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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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 ...
capoocapoo's user avatar
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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 ...
Ivan  Motorin's user avatar
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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 ...
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Bruhat Order on Coxeter Groups

I have been studying Coxeter Groups and started reading on Bruhat Order in the same context. I came across the following definition: Consider $(W,S)$ a Coxeter system with the set of reflections: $T = ...
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parallel walls in Coxeter groups

I am trying to understand the proof of 3.2 in the paper "Coxeter groups are biautomatic" by Osajda and Prytycki [OP]: https://arxiv.org/abs/2206.07804 I will reformulate the statements with ...
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Express reflection w.r.t. combination of vectors as combinations of reflection w.r.t. vectors

Let $\alpha\in\mathbb{R}^m$, then for every $x\in\mathbb{R}^m$ we call reflection of $x$ w.r.t. $\alpha$ the reflection of $x$ w.r.t. the hyperplane $\alpha^\perp=\{y\in\mathbb{R}^m\mid <\alpha,y&...
Giulio Binosi's user avatar
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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 ...
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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 ...
Daniel Vainshtein's user avatar
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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 ...
BulkyMolaMola's user avatar
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Bilinear form associated to Coxeter group

Let me just set the scene before asking my question. Let V be a real vector space with basis $u_1,...,u_n $. Let $W$ be a Coxeter group with generating set $S=\{s_1, ..., s_n \}$ and let $m_{ij}=|...
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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 $T$ ...
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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 ...
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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 =...
Sriotchilism O'Zaic's user avatar
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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}) $ ...
Ian Gershon Teixeira's user avatar
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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 ...
shekh's user avatar
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The closure of the Tits' cone of a Kac-Moody Lie algebra

The question is based on Proposition $5.8$ in 'Infinite dimensional Lie algebras' by Victor G. Kac. The Proposition describes the closure of the tits cone in the $\textit{metric topology}$ of $\...
Irfan's user avatar
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Symmetric group: equation between standard representation and natural matrix representation

I was reading this article and didn't understand their claim. Let $$A = \begin{pmatrix} 1 & 0& 0 &\cdots & 0 \\ -1& 1 & 0 &\cdots & 0\\ 0& -1& 1 &\...
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Relationship between complement right-angled Coxeter groups

Let $\Gamma$ be a graph and $\bar{\Gamma}$ be its complement graph. I am looking for a relationship between the respective right-angled Coxeter groups $G(\Gamma)$ and $G(\bar{\Gamma})$. My intuition ...
return true's user avatar
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Finitely presented finite subgroups of a finitely presented infinite group in GAP?

I am trying to work with infinite Coxeter groups in GAP. An example of such a group would be $\langle a, b, c | a^2, b^2, c^2, (ab)^4, (ac)^2, (bc)^4\rangle$ which can be used to describe the 2D (...
Tom Scruby's user avatar
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Does a permutation determined by set of Inversions?

For $\sigma\in \mathfrak{S}_n$, symmetric group on $\{1,2\dots,n\}$ we denote the set of inversions $\{(i,j) | 1\leq i< j\leq n\text{ and }\sigma(i)> \sigma(j)\}$ by $ \mathcal{I}(\sigma)$. I ...
Learner's user avatar
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Given the Coxeter-Dynkin Diagram of a Lie-Algebra, how do we build the Satake-Tits diagrams of the group's real forms?

For example, $SO(4, \mathbb{C})$ is a 6 complex dimensional complex lie algebra, and its Dynkin diagram looks like $A_2$. It has real forms (6 real dimensions) at least of: $\mathfrak{so}(4) \cong \...
Craig's user avatar
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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 ...
Ian Gershon Teixeira's user avatar
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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 ...
Ian Gershon Teixeira's user avatar
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1 answer
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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 ...
Ian Gershon Teixeira's user avatar
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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 ...
Rob Nicolaides's user avatar
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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 ...
Mathematical Lie's user avatar
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Number of fixed points of generators of reflections (Coxeter) group

Say I have a group with presentation like $$\langle s,t,u \mid s^2,t^2,u^2,(st)^2,(su)^3,(ut)^4\rangle,$$ faithful on set $S$ with exactly one orbit ($|S|$ is known). How could I determine $|\text{Fix}...
Mathematical Lie's user avatar
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217 views

How to count polyhedral rotations?

Suppose I have a regular polytope $P$ which I'm representing as a graph $G_P$ with vertices and edges. I can already put this data into a computer program to find a list of symmetries of $P$---they'...
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Locally finite condition implies finite reflections

I'm looking at the association between reflection groups and Coxeter groups in Bourbaki (books 4-6), and I have a question about some implications of their conditions. Here, $\mathcal{H}$ represents a ...
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Type of the Coxeter matrix for $\tilde{A_n}$

Consider the $(n+1) \times (n+1)$ Coxetermatrix $$ A = \begin{pmatrix} 1 & -\frac{1}{2} & 0 & \cdots & -\frac{1}{2}\\ -\frac{1}{2} & 1 & -\frac{...
Philippe Knecht's user avatar
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Coxeter Groups Classification Proof

I have recently noticed that Bourbaki's proof (Chapter 6, section 4, no.1, Lemma 10, on p205) of the classification of finite Coxeter groups uses the inequality $$ \frac{1}{1+p} + \frac{1}{1+q} + \...
Nerif's user avatar
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Coxeter Groups and representations of $ + $ type

Background: A Coxeter group is generated by reflections, which all have the real eigenvalues $ \pm 1 $, so intuitively it makes sense that all the representations would be real. Certainly every ...
Ian Gershon Teixeira's user avatar
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Are the signed permutation matrices a maximal subgroup of the orthogonal group?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Lift{Lift}$The subgroup of $ \O(n) $ of signed permutations has order $ n!2^n $. It is equal to the ...
Ian Gershon Teixeira's user avatar
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Visual guides request or dictionary for understanding Coxeter-Dynkin diagram

I've seen there are many types of vertices using in Coxeter-Dynkin type diagram to describe the symmetries and polytopes. https://en.wikipedia.org/wiki/List_of_planar_symmetry_groups#Wallpaper_groups ...
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Is there an algorithm to check that a subgroup of a CAT$(0)$ group is *not* quasiconvex?

Let $G$ be a finitely generated CAT$(0)$ group and $H$ a subgroup. If $H$ is quasiconvex then it is finitely generated, so we can immediately conclude that any non-finitely generated subgroup of $G$ ...
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Davis Regular Tessellations of Spheres and Straight Line Coxeter Groups

In Davis' "Geometry and Topology of Coxeter Groups", section B.3, in particular Theorem B.3.1, there is a proof that every finite "straight line" Coxeter group is associated to a ...
Nerif's user avatar
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What is the Coxeter plane?

I'm having trouble in understanding what the Coxeter plane is and how to get graph like this one. Suppose I have a Coxeter group with presentation Suppose I have The Coxeter group $$H_{4}=\left\...
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Why a retraction from a building to an apartment is not isometric?

Let $X$ be an affine building and $\mathcal{A}$ a system of apartment. For any apartment $A\in \mathcal{A}$ and a chamber $C$ in $A$, let us consider the retraction $\rho=\rho_{A,C}\colon X\...
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When does multiplying by an involution increase the Bruhat Order in the Symmetric Group?

Let $w \in \mathrm{Sym}(n)$ for some positive integer $n$. Let $r$ be an involution in $\mathrm{Sym}(n)$, and write it as the product of disjoint transpositions like so: $$r = \prod_{i=1}^k (a_i,b_i) $...
Rob Nicolaides's user avatar
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1 answer
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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, ...
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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) ...
Freeprep's user avatar
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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 ...
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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 ...
user193319's user avatar
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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 ...
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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 ...
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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 ...
Nerif's user avatar
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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 ...
Joppy's user avatar
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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 ...
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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 ...
user193319's user avatar
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Conditions for finiteness of a reflection group

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 ...
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