Questions tagged [coxeter-groups]
For questions about Coxeter groups, an abstract group that admits a formal description in terms of reflections.
229
questions
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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, ...
- 176
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67
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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) ...
- 25
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0
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16
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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 ...
- 7,082
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1
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41
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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 ...
- 7,082
2
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1
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32
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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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1
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31
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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 ...
- 7,082
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0
answers
33
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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 ...
- 76
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45
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Are reflection subgroups corresponding to closed root subsystems always parabolic?
In the third paragraph of this reference, the following is stated: let $W$ be a Coxeter group with set of roots $R$, and let $H$ be a subgroup of $W$ generated by reflections (i.e. by conjugates of ...
- 828
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26
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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 ...
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37
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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 ...
- 53
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1
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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 ...
- 7,082
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1
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53
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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 ...
- 419
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1
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How to find the longest element in a double coset of a Weyl group using SageMath?
I asked a question about computing longest element in a double coset in AskSage
It has not been answered for a long time. So I asked it here.
Let $W$ be a finite Coxeter group. Denote by $W_I$ the ...
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4
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Exercise 5 in Chapter IV, Section 1 on Bourbaki's Lie Groups and Lie Algebras
Let $X$ be a non-empty set and $W$ be a set of permutations of $X$. Assume we are given a set $\mathscr{R}$ of equivalence relations on $X$, a fixed element $x_0\in X$ and a map $\varphi:\mathscr{R}\...
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Defining the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in GAP
I am looking to define the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in the software GAP. In general, one can define these groups separately. For example, let's say
...
- 3,622
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Proof of showing difference between reflections with geometric series
I just started studying the field of reflection and Coxeter groups and I'm trying to prove a result but I'm not sure how to do it. I'll sketch the situation first:
I'd would like to prove that for ...
- 21
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System of representatives in reflection groups and subgroups
I'm working on a paper from Steinberg (1974), "On a theorem of Pittie". The paper is mainly about roots and reflection groups. I'm having trouble understanding the proof of lemma 2.5(a) ...
- 21
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Do mutually commuting reflections have this property in the Bruhat order?
Suppose $(W,S)$ is a Coxeter system and let $<$ denote the (strong) Bruhat Order of $W$; that is $u < b$, there exists some sequence of $t_1,\ldots,t_k \in S^W$ such that $v = ut_1\ldots t_k$ ...
- 165
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2
answers
37
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Generating sets of semi-direct products with $\mathbb{Z}_2$
Suppose a group $G$ splits as a semidirect product $N\rtimes\mathbb{Z}_2$, and let $\phi:G\to\mathbb{Z}_2$ the the associated quotient map. If I have a subset of elements $\{g_1,\dots,g_n,h\}$ of $G$ ...
- 1,211
4
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1
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77
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Discrete groups acting on hyperbolic space
I am interested in groups of isometries which act discretely on uniform spaces of constant curvature, ie $\mathbb{S}^n$, $\mathbb{E}^n$, and $\mathbb{H}^n$. I believe I am right that any such group ...
- 1,211
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1
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22
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Does every reflection generating set of a RA Coxeter group contain a conjugate of every standard generator?
I am interested in understanding generating sets of right-angle Coxeter groups (RACGs) consisting of reflections. More precisely, let $(W,S)$ be a finite rank RACG, and write $R=\{wsw^{-1}\mid s\in S\;...
- 1,211
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23
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Kazhdan-Lusztig polynomial identity
In the book Combinatorics of Coxeter Groups by Björner and Brenti, chapter 5 exercise 13 a) asks the reader to prove the following identity:
$$\sum_{a\in [u,v]}P_{u,a}(q)P_{w_0v,w_0a}(q)=\delta_{u,v},$...
- 161
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1
answer
61
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Hyperbolic Coxeter groups, Humphreys' book
Let $(W,S)$ be an irreducible Coxeter system with non-degenerate bilinear form $B$ on the Euclidean vector space $V$. The simple root attached to $s\in S$ is denoted by $\alpha_s\in V$. Let $\{\...
- 1,502
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43
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A question regarding affine Coxeter groups
Let $\Gamma_n$ denote the isometry group of the regular tessellation of $\mathbb{R}^n$ by $n$-cubes, i.e. $\Gamma_n= \left( \bigoplus\limits_{i=1}^n \mathbb{D}_\infty \right) \rtimes S_n$. Now, let $...
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Generation function for the finite Coxeter group of type $D_k$
Basically I'd like to know how to derive the generation function for the finite Coxeter group of type $D_k$ to be familiar with notes in OEIS A162288:
According to formula section:
'The growth series ...
2
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1
answer
49
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The Bruhat Orders of (finite irreducible) Coxeter Groups as Polytopes
The Strong Bruhat Order of a finite irreducible Coxeter Group satisfies all the axioms of being an abstract polytope. It's also a remarkably nice fact that the Weak Bruhat Order of the Symmetric Group ...
- 165
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1
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Longest element of $D_n$ and the set of positive roots [duplicate]
Personally I am not very familiar with group theory and need some clarifications.
Let's look at $D_n$ and its longest elemements.
According to OEIS A162206 the triangle begins:
$1$;
$1;2;1$;
$1;3;5;6;...
3
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1
answer
36
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What's the classification of Coxeter Groups for which every proper parabolic subgroup is finite?
Let $(W,S)$ be a Coxeter Group. I want to know exactly which Coxeter Groups have the property $\forall J \subsetneqq S $, $W_J$ is finite.
I can think of the finite Coxeter Groups, the Affine Coxeter ...
- 165
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Is there a neat way to construct the Coxeter Group $H_4$ from that of $H_3$ using the fact that $|H_4| = 14400 = 120^2 =|H_3|^2$?
Let $H_4$ and $H_3$ be the usual finite irreducible Coxeter Groups of their names: as presentations $H_4 = \langle s_1,s_2,s_3,s_4\rangle$ subject to the relations $$s_i^2 = (s_is_j)^2 = (s_1s_2)^5 = (...
- 165
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36
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Compute the lengths of Weyl group elements for all possible simple systems
So each choice of simple system for a root system determines a set of simple reflections. The length of an element of weyl group is the minimal number of simple reflections required to express it.
My ...
- 419
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1
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35
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Hyperplane arrangements: Hyperplanes intersecting in a single point
I'm currently reading a text about hyperplane arrangements in a euclidian vector space V. The text says:
[...] without loss of generality, we may assume that some of the
hyperplanes intersect in a ...
- 2,126
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1
answer
41
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Distance labels in regular hyperbolic tilings
Consider the order-4 pentagonal tiling of the hyperbolic plane (shown in the figure Hyperbolic plane tiling with pentagons).
Pick a vertex $s$ (in white), label it with $0$ and then label all the ...
2
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1
answer
79
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Defining a Coxeter group using all reflections
Let $(W, S)$ be a pair of a group $W$ and a subset $S$ consisting of involutions of $W$. We can consider the group $\tilde W$ with presentation $\langle S | \mathcal{R} \rangle$ where $\mathcal{R}$ ...
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How can we define a reflection ordering without the use of roots?
In Björner and Brenti's 'Combinatorics of Coxeter Groups' (pg 137) there is the definition of a reflection ordering for a Coxeter Group: let $(W,S)$ be a Coxeter Group with some induced root system $\...
- 165
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Finding the closest point in a root lattice
Let $L_n$ be a crystallographic root lattice, embedded inside $\mathbb{R}^n$. This means that $L_n$ is the $\mathbb{Z}$-span of the simple roots $\alpha_1, \ldots, \alpha_n \in \mathbb{R}^n$, which ...
- 11.5k
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Reflection subgroups in Coxeter groups
Let $(W, S)$ be a Coxeter system, and set $T = \bigcup_{w \in W} wSw^{-1}$ the set of reflections. A subgroup $R \subseteq W$ is a reflection subgroup if it is generated by a subset of $T$, or ...
- 11.5k
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1
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Coxeter notation for the symmetries of the maximally symmetric unit-distance embedding of $K_{3,3}$ in $\mathbb R^4$
My Shibuya repository now contains unit-distance embeddings in the plane of all cubic symmetric graphs to $120$ vertices, except the first two ($K_4$ and $K_{3,3}$) which do not have this property, as ...
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Which Coxeter Elements have powers that are the longest element of the (Finite, Irreducible) Coxeter Group?
Let $(W,S)$ be a finite, irreducible Coxeter Group. I thought it was true (from Humphrey's book Ex 2 on page 82) that if the Coxeter Number of $W$, $h$, is even then $$c^{h/2} = \omega_0 \quad \dagger$...
- 165
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presentation of a generic algebra
In the book $\ulcorner$Reflection Groups and Coxeter Groups$\lrcorner$ written by J. E. Humphreys, in the beginning of chapter 7 $<$Hecke algebras and Kazhdan-Lusztig polynomials$>$, it defines ...
- 316
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Is there a group with finitely many generators of order 2 that is not a Coxeter group?
Maybe I am missing something obvious here. But in this series of lectures the following is proved to be equivalent under the assumption that the group $W$ has a (finite) set of generators $S$ whose ...
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In the weak Bruhat order, is every element bounded by a power of a Coxeter element?
Let $(W,S)$ be a Coxeter System with $|W| = \infty$. Let $c = s_1\ldots s_{|S|}$ for some total ordering of $S$, a Coxeter element of $W$.
Is it true that for all $w \in W$, there exists a $k\in \...
- 165
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31
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How to find a tessellation from a basis of linear transformations?
Sorry if this is a repeat. It seems like something that should be well known, but I have never found a really good way to write computer code to solve this problem, despite having ad hoc solutions for ...
- 11
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0
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20
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Specific reduced expression of elements in $S_n$ ($W_{B_n}$)
I observed that if we consider elements in $S_n$ as permutations, for example we can write $$\sigma=\begin{pmatrix}
1 &2 &3 &4 \\
4 &3 &2 &1 \\
\end{pmatrix}$$
Then $\sigma$ ...
- 11
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0
answers
81
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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 $...
- 165
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0
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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 ...
- 165
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203
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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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1
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237
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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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1
answer
114
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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 ...
- 677
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1
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139
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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
$$
This is a presentation of a special triangle group $\...
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0
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147
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general properties for Cayley graphs of hyperbolic triangle groups
Is it possible to derive some general properties for Cayley graphs of hyperbolic triangle groups, presented as
$$
\langle a,b,c | a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle \text{, with } \frac1p+\...
- 17.8k