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

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I don't know what this symbol in root systems means (of coxeter groups)

I'm reading Humphreys, Reflection groups and Coxeter groups. The section "Construction of root systems" and the books uses the symbol $ \mathop {\alpha}\limits^{\sim} $ to denote an special element. ...
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Existence of Kazhdan Lusztig basis proof due to Soergel

This question is regarding the proof of the existence and uniqueness of Kazhdan Lusztig basis theorem for an arbitrary coxeter group $W$ due to Kazhdan and Lusztig in his paper "Representations of ...
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22 views

Why is this isomophism of $PGL(2,\mathbb{Z})$ with a Coxeter group injective?

Let $W$ be a Coxeter group with generators $s_1,s_2,s_3$, where $m(s_1,s_2)=3,m(s_1,s_3)=2$, and $m(s_2,s_3)=\infty$. I understand that there's a surjective morphism $\varphi\colon W\to PGL(2,\...
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Is $\operatorname{Stab}(\lambda)$ generated by the simple reflections it contains, for $\lambda\in A_0$?

For a finite Weyl group, the stabilizer of an element in the fundamental domain is generated by the simple reflections of the Weyl group that is contains. Does the same still hold for the closure of ...
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Positive roots expressible as a combination of simple roots whose reflections pairwise commute?

Suppose you have a subset $\alpha_1,\dots,\alpha_r$ of simple roots in a reflection group whose corresponding reflections $\{s_1,\dots,s_r\}$ pairwise commute. Why are the only positive roots ...
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31 views

Proof of a theorem on Reflection Groups

I am reading the book Finite Reflection Groups by Grove and Benson. I didn't understand the following proof. See $(a_1,t)$. What is $t$ here? Then Why the inequality $(r,t)-2(r,r_{i_1})(r_{i_1},t)&...
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Finding a polytope in the Cartan Subalgebra

The finite Coxeter groups can be realized as symmetry groups of (semi)-regular polytopes. Not all semi-regular polytopes can be realized this way, but all regular polytopes can. Some examples of ...
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32 views

Name of a family of Coxeter groups

From the following image I know that the first of group is the symmetric group of rank $n$ and the second is known as the Hyperoctahedral group. I want to know if someone knows the name of the ...
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15 views

Number of conjugacy classes or bounds for the index of stabilizers on finite Coxeter groups.

For a finite group $G$ denote by $\operatorname{conj}\left(G\right)$ the number of conjugacy classes of $G$ and let $q\left(G\right)$ be the quotient: \begin{align} q\left(G\right)=\frac{\...
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1answer
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Why is reflection length equal to codimension of fixed subspace in a real reflection group?

If you have a finite, real reflection group, why can the length function $\ell(-)$ be interpreted as the codimension of the fixed subspace, or alternatively, as the number of eigenvalues different ...
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Kazhdan-Lusztig polynomials for the longest element in finite Coxeter groups

Suppose $W$ is a finite Coxeter group and that $w_\circ$ is the longest element. I want to prove that $P_{x,w_\circ}=1$ for all $x \in W$. I know from Corollary 7.14 in the book Humphreys p. 167 ...
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Looking for conditions on generator sets of the form $\{a,b \}$ and $\{a,b,c\}$ on the group $\mathbb{Z}_2^3 \rtimes S_3$

Let $G$ be the group $\mathbb{Z}_2^3 \rtimes S_3$ with the natural action of $S_3$ on the coordinates of $\mathbb{Z}_2^3$. I want to know if there are subsets of $G$ of two elements or also 3 elements ...
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14 views

Connceted componets of a complement of a union of hyperplanes

Let $(W,S)$ be an irreducible coxeter system with graph $\Gamma$ and associated bilinear form $B$. a) $B$ is nondegenerate, but not positive definite. b) For each $s \in S$, the coxeter graph ...
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Hyperbolic coxeter groups proof of a proposition

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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20 views

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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23 views

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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6 views

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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42 views

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 $s^2_i=e$, $(s_is_j)^{m_{...
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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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34 views

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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37 views

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, n-1)(n-1,n-2)\cdots(21)(n-1,n-2)...
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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, (s_1s_2)^4=(s_is_{i+1})^3=(s_is_j)^...
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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
82 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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95 views

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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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 $m_{i,j}=m_{j,i}$...
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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: $$T(w)f(Y)=\...
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123 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
43 views

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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297 views

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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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 ...