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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presentation of a generic algebra

In the book $\ulcorner$Reflection Groups and Coxeter Groups$\lrcorner$ written by J. E. Humphreys, in the beginning of chapter 7 $<$Hecke algebras and Kazhdan-Lusztig polynomials$>$, it defines ...
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51 views

Is there a group with finitely many generators of order 2 that is not a Coxeter group?

Maybe I am missing something obvious here. But in this series of lectures the following is proved to be equivalent under the assumption that the group $W$ has a (finite) set of generators $S$ whose ...
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27 views

In the weak Bruhat order, is every element bounded by a power of a Coxeter element?

Let $(W,S)$ be a Coxeter System with $|W| = \infty$. Let $c = s_1\ldots s_{|S|}$ for some total ordering of $S$, a Coxeter element of $W$. Is it true that for all $w \in W$, there exists a $k\in \...
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26 views

How to find a tessellation from a basis of linear transformations?

Sorry if this is a repeat. It seems like something that should be well known, but I have never found a really good way to write computer code to solve this problem, despite having ad hoc solutions for ...
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24 views

How to prove that the root systems of Coxeter groups are discrete?

First I want to know the definition of "discrete" here. I guess it can be inferred from the bilinear form on the Coxeter datum. But it is still hard to organize the prove by finding the ...
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15 views

Specific reduced expression of elements in $S_n$ ($W_{B_n}$)

I observed that if we consider elements in $S_n$ as permutations, for example we can write $$\sigma=\begin{pmatrix} 1 &2 &3 &4 \\ 4 &3 &2 &1 \\ \end{pmatrix}$$ Then $\sigma$ ...
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35 views

Dot Product of Vectors of Roots of Unity

Let $n \in \mathbb{N}$ and $i$ be the usual complex number such that $i^2=-1$. Let $\zeta = \exp(\frac{\pi}{n} i)$, $v = [1,\zeta,\zeta^2,\ldots,\zeta^{n-1}]$. Given $J \subseteq \{1,\ldots,n\}$, let $...
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28 views

When is the subgroup product of two parabolic subgroups of a Coxeter Group, the Coxeter Group itself?

Let $W$ be a Coxeter Group generated by simple reflections $S$. If $I,J\subseteq S$ and $W_I = \langle s | s \in I \rangle$ when is it true that $W_IW_J = W$? I am secretly hoping that the answer ...
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189 views

How to find generators of translation subgroup of an abstract reflection (coxeter) group

I have an infinite reflection group https://en.wikipedia.org/wiki/Coxeter_group Take for example the affine groups $[4,4],[4,3,4],[4,3,3,4]$... I'd like to get an explicit expression for generators ...
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16 views

How to project a regular n-hypercube onto a Petrie polygon and is natural there a 3D analogue?

Given the $n$-hypercube who's vertices correspond to the $n$ by 1 vectors with entries in {-1,1}, is there a general construction for a 2 by $n$ matrix, $M$, such that $M.v$ gives the projection of ...
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124 views

Which root systems admit a proper root subsystem with full span?

Let $\Phi$ be an irreducible root system of $\mathrm{rank}(\Phi) = n$. I am allowing the case where $\Phi$ is not reduced. Say that a subset $\Psi \subset \Phi$ is a root subsystem if it is a root ...
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91 views

How can one recognize when two Coxeter diagrams represent the same uniform polytope?

If we enumerate uniform polytopes generated by Coxeter diagrams, we find many cases where two apparently distinct Coxeter diagrams yield the same uniform polytope. A small handful of examples: the ...
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133 views

Does $(xzy)^{s/2}$ preserve some special kind of property like orientation?

Let's look at the following presentation: $$ \Delta^*(p,q,r;s/2)=\langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=((abc)^2)^{s/2}=1\rangle $$ This is a presentation of a special triangle group $\...
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101 views

general properties for Cayley graphs of hyperbolic triangle groups

Is it possible to derive some general properties for Cayley graphs of hyperbolic triangle groups, presented as $$ \langle a,b,c | a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle \text{, with } \frac1p+\...
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1answer
51 views

Embedding of the 1-skeleton of a Coxeter group into its Davis complex

Let $(W,S)$ be a Coxeter system and let $\Sigma$ be the corresponding Davis complex. It is well-known that the Davis complex may be equipped with a piecewise Euclidean metric so that it is a proper, ...
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38 views

Reference for “how to read of the faces of a uniform polytope from its Coxeter diagram”

It appears as if the (combinatorial type of the) faces of (Wythoffian) uniform polytopes can be read off from their Coxeter-Dynkin diagram by deleting appropriate vertices. I think I understood how ...
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1answer
124 views

Crystallographic root system Coxeter Groups

In Humphreys book in "Reflection Groups and Coxeter Groups" he defines a root system $\Phi$ to be crystallographic if it satisfies $\frac{2(\alpha, \beta)}{(\beta, \beta)} \in \mathbb{Z}$ $(\star)$ ...
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34 views

Partial Order on a Vector space with a Crystallographic Coxeter Group

Let $V$ be a vector space, and $W$ an irreducible, crystallographic Coxeter group on $V$ with simple root system $\Pi = \{a_1 \cdots a_n \}$. We define a partial ordering on $V$ by $u \geq v$ iff $u - ...
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In a Coxeter system, does $I^w\cap J=I^{w'w}\cap J$ whenever $w'\in W_I$?

Suppose $(W,S)$ is a Coxeter system, and $I,J\subseteq S$. Let $I^w=w^{-1}Iw$ denote the conjugate of $I$ under an element of $W$. If $w'\in W_I$, the parabolic subgroup for $I$, is it true that $I^{w'...
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1answer
38 views

Orbits of pairs of multi-indices under the diagonal action of the symmetric group

This question concerns a statement made on page 168 of Dipper, R. and Donkin, S., 1991. Quantum GLn. Proceedings of the London Mathematical Society, 3(1), pp.165-211. I have tried to include all ...
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65 views

Question about a certain involution on a Coxeter group $W$.

This is a small question that arose while reading the paper On Okuyama's Theorems About Alvis-Curtis Duality by M. Cabanes. It can be read here. Let $(W,S)$ be a finite Coxeter system, $l$ the length ...
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27 views

Proving the Deletion Condition using the Strong Exchange Condition

I wish to prove the following statement. Let $w=s_{r_1}\cdots \; s_{r_k}$ with $\ell (w) < k$. Then there exist $i < j$ such that $w=s_{r_1}\cdots \; \hat{s}_{r_i}\cdots \; \hat{s}_{r_j}\...
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2answers
89 views

What are the word hyperbolic affine Coxeter groups?

It is well-knwon that all affine (irreducible) Coxeter systems can be classified by their Coxeter graphs, see Wikipedia. The corresponding diagrams are $(\tilde{A}_n)_{n \geq 1}$, $(\tilde{B}_n)_{n \...
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100 views

Let $w_0$ be the element of longest length in a Coxeter group. Show that $l(w_0w)=l(ww_0)=l(w_0)-l(w)$? Find $w_0$ explicitly in $S_n$.

Let $w_0$ be the unique longest element in $W=S_n$. Let us show that $$l(ww_0)=l(w_0)-l(w)$$ for any $w \in W$. We proceed by induction on $l(w)$. First, let $S$ be the generating set for $W$. In ...
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27 views

Does every finite Coxeter group have a unique longest element?

If $W$ is a finite Coxeter group, then it has a unique longest element $w_0$. In particular, $w_0$ is an involution. This is a (paraphrased) quote from Fayers' paper $0$-Hecke Algebras of finite ...
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32 views

There is an element of every possible length in $[W_{\theta} \backslash W]$

Let $(W,S)$ be the Weyl group of a root system with base $\Delta$, and let $\theta \subset \Delta$. Let $W_{\theta}$ be the group generated by $\theta$. It is a general result that in every right ...
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53 views

Coxeter graph of the group $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$

I am reading the first chapter of Combinatorics of Coxeter Groups by A.Björner and F.Brenti. In the first example they say that the graph with $n$ isolated vertices (no edges) is the Coxeter graph of ...
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When is the group action transitive?

Let $W$ be a finite reflection group with irreducible root system $\Phi$. Let $\Delta$ be the set consisting of reflections $s_{\alpha}$ corresponding to pairs $\{\alpha, -\alpha \}$ of roots in $\...
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227 views

What sort of groups are generated by a single conjugacy class?

To clarify, I am not looking for a classification but rather for well-researched examples of families of (finitely generated) groups generated by a single conjugacy class. A collection of examples, ...
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39 views

Coxeter exchange condition in symmetric group

I would like to prove (for purposes of illustration mainly) that the symmetric group $S_n$ with the set $S$ of adjacent transpositions $(i, i+1)$ is a Coxeter group by proving that it satisfies the ...
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Intersection of Levi subgroups via intersection of their Weyl groups

Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a maximal torus $T\subset G$. Let $M,L \subset G$ be its Levi subgroups containing $T$ (note that we do note assume that $M,L$ are ...
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1answer
124 views

Motivation for definition of bilinear form in linear representation of Coxeter groups?

In a set of notes on Coxeter groups I am reading the following definitions are made: Let $M = (m_{ij})_{1 \leq i,j \leq n}$ be a symmetric $n \times n$ matrix with entries from $\mathbb{N} \cup \...
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3answers
148 views

Symmetries of a 4-demicube

I do not really understand, what is the symmetry group of a 4-demicube. I've read that the 4-demicube coincides with the 16-cell. In what sense? I can see that the corresponding graph (i.e. taking ...
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1answer
111 views

Free presentations of Weyl groups

Can someone please provide me with a reference for the fact that the Weyl group $W$ associated to a root system $\Phi$ can be realised as a Coxeter group? This means that a Weyl group $W$ has a ...
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1answer
39 views

Is every convex fundamental region regularly open?

Recall that a regular open set is an open set $U$ for which $U = \operatorname{int} \operatorname{cl} U$. A fundamental region for a finite group $G\leq \mathcal{O}(V)$ is an open set $F\subseteq V$ ...
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107 views

Reduced Expression for Reflection in Weyl Group

Let $W$ be a Weyl group of a semisimple Lie algebra $\mathfrak{g}$. Let $\beta$ be a positive root, $\alpha$ a simple root such that $\beta=w(\alpha)$. It is a straightforward fact that we can ...
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How would one classify point groups?

By point group I mean a finit subgroup of $\mathrm O(\Bbb R^n)$. Lists of point groups for some small dimensions are found on Wikipedia, but I am not certain about their completeness. As there seem ...
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Counting inversions of random elements in coxeter groups

I am trying to find a general interperetation to the following facts (pls be patient to read it). Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of ...
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Embeddings of Coxeter Groups of Rank $3$ into $\text{SO}(3)$

Let $W = \langle x_1, x_2, x_3 \;|\; (x_ix_j)^{m_{ij}} \rangle$ be an irreducible Coxeter group, i.e., the graph with vertices $v_1, v_2, v_3$ and edges between all pairs $(v_i, v_j)$ with $m_{ij} \ne ...
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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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1answer
52 views

Can every positive root of a Coxeter group be written as a simple root and a positive root?

Can every positive root of a Coxeter group be written as a simple root and a positive root? I think that this is possible. For example, in type $B_2$, the set of positive roots are $\alpha_1, \alpha_2,...
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Can generators of a Coxeter group be redundant?

I’m just beginning to learn about Coxeter groups and at one point the textbook’s author seems to take it for grant that each generator cannot be “reduced” into a product of other generators. ...
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199 views

About parabolic subgroup of a Weyl group

Let $W$ be a Weyl group/Coxeter group. Let $\Phi$ be the associated root system, fix a positive root system $\Phi^+$ and let $\Delta$ be the set of simple roots. Let $W_I$ be the parabolic subgroup ...
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67 views

Reflection length of the longest word in a Coxeter group.

Let $(W,S)$ be a Coxeter system. The set of reflections of $W$ is $S=\{wsw^{-1}:s \in S, w \in W\}$. The reflection length of $w \in W$ is defined as the minimal number $m$ such that $w = r_1 \cdots ...
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122 views

How to show bilinearity

Given a Coxeter group $\varGamma =\langle \rho_0, \rho_1, \ldots, \rho_{n - 1}\rangle$ which at least satisfy the relation $(\rho_i\rho_j)^{p_{ij}}=1, \ \ 0\leq i, j \leq n - 1,$ where $p_{ii}=1$ and $...
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101 views

Type B Catalan numbers as signed permutations

The Catalan numbers are in bijection with the 123, 132, etc. avoiding permutations in $S_n$. If we move to type B, the type B Catalan numbers is $\binom{2n}{n}$, and the permutation group is the ...
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93 views

Coxeter length in the symmetric group equals number of inversions

Let $S_n$ be the symmetric group on the set $\{1,\dots, n\}$ and $\sigma\in S_n$. An inversion of $\sigma$ is a pair $(i,j)$ such that $1\leq i<j\leq n$ and $\sigma(i)>\sigma(j)$. Let $S_n$ be ...
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183 views

Why $(1+x)(1+x+x^2)\cdots(1+x+x^2+\cdots+x^n)$ gives the Poincare series?

I am looking for an explanation of the fact that the polynomial $$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$ with Mahonian numbers gives the Poincare series for the symmetric group $S_{n+1}$ considered as a ...
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1answer
15 views

$B$-adapted pairs and $\hat{G}$ acting on $W$

Let $(G,B,N,S)$ be a Tits system, and let $\varphi: G \rightarrow \hat{G}$ be a $B$-adapted homomorphism. This means the kernel of $\varphi$ is contained in $G$, and for every $g \in \hat{G}$, there ...
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59 views

In a Bruhat-Tits building, $N$ is the stabilizer of the standard apartment

Let $(G,B,N,S)$ be a Tits system. Assume the system is saturated, which is to say that if we define $$H = \bigcap\limits_{n \in N} nBn^{-1}$$ then $H \subseteq N$. To an affine root system whose ...