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

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Hyperbolic Coxeter Systems

This question is regarding the proof of a proposition in the text book reflection groups and coxeter groups by Hymphreys section 6.8. Prop: Let $(W,S)$ be an irreducible coxeter system with graph ...
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signature of a bilinear form

This question is regarding the proof of a lemma in the book Reflection groups and Coxeter groups by Humphreys section 6.8. Lemma: let $E$ be an n-dimensional real vector space endowed with a ...
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Geometric interpretation of length function of a coxeter group

It is about an exercise in Humphrey's Reflection groups and Coxeter groups exercise 1 section 5.6. Let (W,S) be a Coxeter system. It is assumed throughout the chapter that S is finite. Let $\sigma : ...
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Order of parabolic subgroups of affine Weyl groups

I have a question about computing the order of an arbitrary parabolic subgroup of an affine Weyl group $W_a$. Given a proper subset $I \subset S_a$ associated with the reflections for the fundamental ...
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How does the Weyl group act on the root system of type $B_n$?

Suppose $\{e_1,\dots,e_n\}$ are the standard unit vectors in $\mathbb{R}^n$. Then the root system of type $B_n$ consists of $\pm e_i$, and $\pm(e_i\pm e_j)$ for $i\neq j$. I know the Weyl group $W$ ...
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Lenght of the affine transformation $s_{\varphi_n, \, 1} \cdot \dots \cdot s_{\varphi_1, \, 1}$

Let $\{\varphi_1, \, \dots , \, \varphi_n\}$ be a subset of positive roots of a root system $\varPhi$ and consider the affine reflections $\{s_{\varphi_n, \, 1}, \, \dots , \, s_{\varphi_1, \, 1} \}$ ...
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Is a Coxeter group W operating on a finite set X also finite?

Please regard this A COMBINATORIAL CONSTRUCTION FOR SIMPLY–LACED LIE ALGEBRAS on page 7 (it is brief but I hope that page is enought introduction on that topic). Can I argue, and if so how, that the ...
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In a reflection group, the longest word $w_0$ contains all simple reflections

This is Exercise 2 of section 1.8 in Humphreys' "Reflection groups and Coxeter groups", p.16. The longest word $w_0$ in a finite reflection group $W$ acting on a Euclidean space (with a specified ...
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Relation between reflection group and coxeter group

Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group. An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, ...
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If $P_I$ is a parabolic subgroup in $GL_n(k)$, is $P_I$ conjugate to ${P_I}^-$ under some element of $W$?

Suppose $P_I$ is a standard parabolic subgroup of $GL_n(k)$ (we can assume $k$ is finite, but I doubt it matters). Is $P_I$ conjugate to the opposite parabolic ${P_I}^-$ under some element of the Weyl ...
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Is there a quick way to find the longest element of a subgroup of a Weyl group?

Suppose $W$ is a Weyl group generated by a set of reflections $S$. For $I\subset S$, there are subgroups $W_I$ generated by the reflections $s\in I$. If $w_I$ denotes the longest element of $W_I$, is ...
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What is the longest element of $S_n$ as a product of adjacent transpositions?

I can't seem to get this to work. According to wikipeda, the longest element of $S_n$ should be expressible as a product of $n(n-1)/2$ adjacent transpositions by $$ (n, ...
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On groups with presentations $ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $…

$$ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. What about the following ...
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Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$…

Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_d+A_b+A_d$ (where $d$ is dark, damn...). It's ...
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Intersection of halfspaces and hyperplanes

If $H_1$, $H_2$ and $H_3$ are hyperplanes in an $n$ dimensional vector space $V$ then I want to prove that the linear span of $ H_1\cap H_2^+\cap H_3$ is $H_1\cap H_3$. Clearly the span is contained ...
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Coxeter presentation of Hyperoctahedral group

I know that the hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$ has the presentation $\langle s_{\text{1}},\ldots,s_n:s_{\text{1}}^{\text{2}}=s_i^2=1, ...
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Generating set of Coxeter group from a $(B,N)$-pair?

Suppose we have a $(B,N)$-pair for a group $G$. The group $W:=N/(B\cap N)$ is known to be generated by a set $S$ of involutions. Apparently $$ S=\{w\in W:B\cup BwB\text{ is a group}\}. $$ The reason ...
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1answer
80 views

A subset of roots whose mutual angles agree with those of a simple system

I would appreciate help/hints solving the following exercise from Humphreys book "Reflection Groups and Coxeter Groups", page 11, exercise 1. Let $\Phi$ be a root system of rank $n$ of unit ...
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Every positive system of roots contains a unique simple system.

The following question is in effort to understand a proof to a theorem appearing in "Reflection Groups and Coxeter Groups" by Humphreys on page 8. Let $\Phi$ be a root system in the euclidean space ...
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Lie algebra-like structure corresponding to noncrystallographic root systems

In the classification of Coxeter groups, or equivalently root systems: $$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$ with $p \geq 7$, the last four fail to generate any ...
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A question about Coxeter groups.

Let $G$ be a group, and $x,y,z \in G - \{1\}$, where $1$ is the identity element of $G$. Assume that $x^2=y^2=z^2=1$, $xy =yz$ and $xz$ is of infinite order. Can $G$ be a Coxeter group? Can $G$ be ...
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When is a right-angled Coxeter group one-ended?

Let $\Gamma$ be a simplicial graph (ie. without multiple edes nor loops). We define the associated right-angled Artin group $A(\Gamma)$ by the presentation $$\langle v \in V(\Gamma) \mid [u,v]=1 \ ...
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Reduce expresion

How can I reduce this the following expresion $$\sum_{k = 1}^m \frac{x_1 \sin \left( \frac{2k\pi}{m} \right) - x_2 \cos \left( \frac{2k\pi}{m} \right)}{x_1 \cos \left( \frac{2k\pi}{m} \right) + x_2 ...
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Centralizers of reflections in parabolic subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with ...
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When is a stabilizer group a reflection group?

Let $G$ be a compact, connected Lie group and $K$ a closed, connected subgroup. If $K = T$ is a maximal torus, it is well known that $W := N_G(T)/Z_G(T) = N_G(T)/T$ is a finite reflection group, the ...
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What decides if a Coxeter Group is “crystalline” or “non-crystalline”?

I am currently writing a research essay concerning Crystallographic theories applied to Virology. Do keep in mind that I am not a mathematician and I know very little about Coxeter groups in general. ...
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The product of Weyl groups is the Weyl group of the product

Let $G$ and $H$ be compact, connected Lie groups. Write $W(-)$ for the Weyl group. Then it is not hard to see that $$W(G \times H) \cong W(G) \times W(H).$$ Where can I find a published statement of ...
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An injection of Weyl groups

I've shown, quite accidentally, that Weyl group of $F_4$ injects into the Weyl group of $E_6$ as the subgroup of elements normalizing a maximal torus $T^4$ of $F_4$. One might a priori expect other ...
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Affine Hecke algebras and Lusztig relations

I study the book "Affine Hecke algebras and orthogonal polynomials" by I.G. Macdonald. He propose a formula in section $4.2$, especially formula $(4.2.9)$. This formula is the following: ...
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111 views

Weyl groups: correspondence of reflections and roots?

If W is the Weyl group of some ADE-type Lie algebra, and w is an element corresponding to a reflection (not just an involution), does it necessarily correspond to a root?
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Isomorphic Coxeter Groups

Is it true that two Coxeter groups having the same Coxeter matrix (equivalently, the same Coxeter graph) isomorphic? Because otherwise, the definition of a Coxeter group from its Coxeter matrix does ...
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1answer
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Use GAP3 program for computing the h-polynomials of Lusztig

I am using GAP to compute things in Kazhdan-Lusztig theory, especially using the package "Chevie". According to the GAP3 Manual, the base change between the "C"-basis and "T"-basis can be computed, ...
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Texts on Coxeter groups

I'm looking for an introductory text on Coxeter groups. It can assume undegraduate knowledge of Algebra (Groups up to and including the Sylow theorems in Fraleigh, elementary knowledge of rings, ...
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Lusztig's $h$-function of a dihedral group

Following the notations in Hecke algebras with unequal parameters, let $(W,S,L)$ be a weighted Coxeter system, and $H$ be the corresponding Hecke algebra with $\{c_w |w \in W\}$ the Kazhdan-Lusztig ...
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Create Platonic solids from the coxeter group (vertexes & edges & faces)

How can one define vertexes, edges and faces from the Coxeter group? For example, for all platonic solids? I would like to create a general function that takes the Coxeter diagram as input, and gives ...
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39 views

Simple systems are the “smallest”

Let $\Phi$ be a root system of a finite reflection group $W$. Let $\Delta$ be a simple system in $\Phi$. I want to prove that $\Delta$ is "the smallest" set which generates $W$, more precisely: There ...
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Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
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93 views

Is every finitely generated Kleinian group commensurable to a Coxeter group?

Or, to a finitely generated reflection group? Here, I do not insist that the Coxeter group is represented as a hyperbolic reflection group. If not, what is an counterexample? And what is a ...
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Embed one Coxeter System into another

What is a good reference that explains all the braid relations and diagrams for Coxeter systems concisely? In particular, how do I embed $H_3$ inside $D_6$, or $H_4$ inside $E_8$? Any hints?
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No cycles in finite coxeter graphs

Is there an elementary (no consideration of root systems involved) proof of the fact that the graph of an finite coxeter system doesn't entail any cycle? I got as far as this: If there were any cycle ...
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Right-angled Artin groups are residually finite

I know that residual finitness of RAAGs (Right-Angled Artin Groups) follows from linearity, but does there exist a more direct proof, maybe simpler? EDIT: I added a proof based on cube complexes ...
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Representation of cell location in hyperbolic plane

I want to represent an order-5 square tiling (image from Wikipedia; more text below image): Obviously for a simple grid I can uniquely refer to a given square by its ...
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A question on universal Coxeter group

In the set up of Lusztig's Hecke algebra with unequal parameters, let $W$ be a universal Coxeter group with finite many simple reflections, that is, $W=\langle s_i,i=1,2,\cdots,n | s_i^2 =1\rangle$. ...
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coxeter graph and root system

I want to show that a coxeter graph $\Gamma$ is connected if and only if its root system $\Phi$ is irreducible. So let $\Delta$ be a simple system of $\Phi$, and $\Delta$ is also our simple system. ...
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angles between simple roots are obtuse, problem with proof

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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125 views

Angle between roots in a root system

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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105 views

Isn't the picture on Wikipedia about Weyl Chambers wrong?

Wikipedia's article on Weyl groups shows an example of a root system and the corresponding fundamental chambers (in my understanding, also known as fundamental regions or fundamental domains). In this ...
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Subspace invariant under irreducible Coxeter group

I'm trying to show that if $G$ is an irreducible Coxeter group, then it acts irreducibly on vector space $V$. That is, $V$ has no nontrivial $G$-invariant subspaces. I started by assuming that $V$ has ...
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60 views

Proof about coxeter groups

Prove: If a coxeter system $(W,S)$ is reducible, then it is the product of parabolic subgroups. Reducible system means the coxeter diagram is disconnected. Parabolic subgroup: let $S$ be the set ...
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What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...