Questions tagged [coxeter-groups]

A Coxeter group is a group defined by a presentation by involutions $r_i$ with relators $(r_ir_j)^{m_{ij}}=1$ for certain family $(m_{ij})$ of integers greater than 1.

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Chevalley-Solomon formula and Weyl character formula

Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
Antoine Labelle's user avatar
4 votes
1 answer
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For which quadratic number field, the algebraic integers are cusps for some Coxeter group?

Let $H^2=\{(x,y)\mid y>0\}$ be the hyperbolic upper-half plane. Let $K=Q(\sqrt{d})$ be a quadratic number field, and $\mathcal{O}_K$ be the ring of algebraic integers in it. Let $\Gamma=\Delta(p,q,...
zemora's user avatar
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8 votes
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Inversions for parity preserving presentations

I've gotten stuck on a slightly random combinatorial question, and I'm doing a bit of a shot in the dark here to see if someone else has thoughts about it. I'm interested in studying a permutation of ...
Ben Webster's user avatar
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When is an affine left cell finite?

Consider an affine Weyl group $\hat W$ of a simple Lie type. Let $w \in \hat W$ and let $C^L(w)$ denote the left cell in $\hat W$ containing $w$. Is there a good criterion to test whether $C^L(w)$ has ...
Qixian Zhao's user avatar
3 votes
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92 views

Does there exist a finite-volume hyperbolic Coxeter polytope with these properties?

I searched for a finite-volume, hyperbolic Coxeter polytope of dimension $n \geq 4$ with the following properties $a$ and $b$. $a$) It has exactly one ideal vertex; $b$) if a bounded facet and an ...
Edoardo Rizzi's user avatar
5 votes
1 answer
184 views

What is the effect of tensoring with the sign representation on irreducible modules for a Type D Weyl group?

Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers. The Weyl ...
Christopher Drupieski's user avatar
2 votes
1 answer
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When are these irreducible complex representations for the Type D Weyl group self-dual?

Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers. The Weyl ...
Christopher Drupieski's user avatar
15 votes
1 answer
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Is this a typo in Macdonald's paper "The Poincaré Series of a Coxeter Group"?

I have a question about the proof of lemma 2.14 in Macdonald's paper The Poincaré Series of a Coxeter Group [1], where he used induction on $l(w)$ to prove that if $|E|=|R(w)|$, then $E=R(w)$. The ...
zemora's user avatar
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Combinatorics behind certain induction of characters of the Coxeter group of type $B_n$

Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as ...
Suzet's user avatar
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1 answer
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Minimal dominant permutation in weak order

Consider $S_\infty$ as a Coxeter group with Coxeter generators the adjacent transpositions $s_i$, $i\geq 1$. We view elements of $S_\infty$ as functions $u:\mathbb{N}\to\mathbb{N}$. Recall the Lehmer ...
Matt Samuel's user avatar
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3 votes
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290 views

Is G(4,7) a Coxeter group

Let $G(4, 7)$ be an abstract group with the presentation $$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$ Richard Schwartz considered ...
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Question on the Tits cone of an irreducible affine Coxeter group

Let $(W,S)$ be an irreducible affine Coxeter group, $M=(m_{ij})$ be its Coxeter matrix, and $\{\alpha_s\}_{s\in S}\in V$ be the system of simple roots in the standard geometric realization, so the $\{\...
zemora's user avatar
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Weyl groups are Coxeter groups proof

I'm reading part of a proof that says that Weyl groups of apartments of buildings are Coxeter groups. Let $\Delta$ be a building and let $\Sigma$ be a fixed apartment of $\Delta$. Let $C$ be a fixed ...
Anonmath101's user avatar
0 votes
1 answer
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A question on irreducible affine Coxeter groups

I have a question about affine Coxeter groups when reading Humphreys's book: Let $(W,S)$ be an irreducible affine Coxeter group, $M=(m_{ij})$ be its Coxeter matrix, and $\{\alpha_s\}_{s\in S}\in V$ be ...
zemora's user avatar
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A basis for the 0-Hecke ring

Let $(W,S)$ be a Coxeter system of type $A_n$, with $$S=\{s_1,\ldots,s_n\}$$ satisfying the usual relations, and let $R=\mathbb{Z}[x_1,\ldots,x_{n+1}]$ be a polynomial ring. $W$ acts on $R$ by ...
Matt Samuel's user avatar
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1 vote
1 answer
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Linear independence of reciprocals of products of closed sets of roots in type $A$ inversion sets

Consider the root system $R$ for a Coxeter system $(W,S)$ of type $A_n$ with a choice of simple roots. Denote by $I(w)$ for $w\in W$ the set of positive roots $\beta\in R^+$ such that $w(\beta)$ is a ...
Matt Samuel's user avatar
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Rank of Coxeter matrix

Let $Q$ be a quiver and $\Phi_Q$ be the Coxeter matrix of $Q$. Then $\Phi_Q\pm I$ are full-rank?
user145752's user avatar
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Writing an element of a free product of $C_2$'s as a product of order-$2$ elements

My question is simple: Suppose that $G$ is isomorphic to the free product of finitely many copies of $C_2$. Is it true that any element $g \in G$ can be written as a product $g = s_1 \dotsm s_m$ such ...
Jeff Yelton's user avatar
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2 votes
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110 views

Do Weyl groups generate the exceptional Lie groups as sequences of reflexions in the Weyl chambers?

Platonic groups of symmetry are Weyl groups for the exceptional Lie algebra E6->E8, as root systems. These can be viewed as mirrors in a kaleidoscope (Goodman). I would like to know if one can ...
Lucian Ionescu's user avatar
8 votes
2 answers
261 views

One element commutation classes of reduced decompositions of the longest element of the Weyl group

For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the ...
Didier de Montblazon's user avatar
7 votes
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245 views

Generating cycles inside Tits' graph of words for a positive braid

Let $Br_n$ be the braid group and consider words in its generators (not in the inverses). Two such words define the same "positive" braid if one can be obtained from the other by commuting ...
Allen Knutson's user avatar
7 votes
1 answer
533 views

Signed permutations and $ \operatorname{SO}(n) $

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Lift{Lift}$The subgroup of $ \SO(n) $ of determinant-$1$ signed permutations has order $ n!2^{n-1} $...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
341 views

Parabolic subgroup of Weyl group

Let $W$ be the Weyl group of a semisimple algebraic group $G$. $I$ be the simple roots. $J\subset I$ generate a parabolic subgroup of $W$ denote by $W_J$. $w^J$ is the shortest representative of $w$ ...
fool rabbit's user avatar
8 votes
1 answer
395 views

When are groups generated by reflections in a triangle discrete?

Take a triangle in the (Euclidean or hyperbolic) plane, and consider the group of isometries generated by the reflections in the three sides of the triangle. If the angles between adjacent sides are ...
Ethan Dlugie's user avatar
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3 votes
1 answer
211 views

Eriksson's thesis "Strongly convergent games and Coxeter groups"

The diamond lemma has recently come up in my teaching, and as always I've been looking for nice and simple applications. This has reminded me of the thesis Kimmo Eriksson, Strongly convergent games ...
darij grinberg's user avatar
2 votes
1 answer
263 views

Coxeter subgroups of Coxeter groups vs Artin subgroups of Artin groups

Given two Coxeter groups $W(\Gamma)$ and $W(\gamma)$ of equal rank and their Artin groups $A(\Gamma)$ and $A(\gamma)$, is $W(\Gamma)\supset W(\gamma)$ (as reflection groups) equivalent to $A(\Gamma)\...
Daniel Sebald's user avatar
5 votes
2 answers
257 views

Reflection quotients of Coxeter groups

I am interested in a concept somehow dual to reflection subgroups. A reflection quotient of a Coxeter system $(W, S)$ shall be a surjective homomorphism $W \to W'$ to a Coxeter group $W'$ such that ...
Levi Ryffel's user avatar
4 votes
0 answers
207 views

Group generated by Coxeter groups of $B_8$ and $E_8$

Note: $B_8$, $D_8$, and $E_8$ are referring to the Coxeter groups. What is the smallest group $G$ with the following properties? $G$ has a subgroup isomorphic to $E_8$. $G$ has a subgroup isomorphic ...
Daniel Sebald's user avatar
3 votes
1 answer
210 views

Finiteness of a reflection group

Suppose that $V$ is a finite-dimensional real vector space and that $W\subseteq \operatorname{GL}(V)$ is a subgroup generated by reflections (elements $s$ of order $2$ whose locus of fixed points $H_s$...
inkspot's user avatar
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3 votes
0 answers
114 views

Why inherit the Tits systems structure by a $B$-adapted homomorphism?

Let $(G,B,N,S)$ be a Tits system and $\phi\colon G\longrightarrow \hat{G}$ a $B$-adapted in the sense of the paper Groupes réductifs sur un corps local: I of Bruhat–Tits. They said that $\phi$ is a $B$...
M masa's user avatar
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4 votes
0 answers
141 views

Polynomial invariants of infinite reflection groups

It is a famous theorem of Shepard-Todd that the ring of invariant polynomials of a finite complex reflection group $W$ acting on a complex vector space $V$ is actually itself a polynomial ring. In ...
K Goldman's user avatar
4 votes
0 answers
179 views

The Weyl group of Kac-Moody algebra and Coxeter group

Let $\mathfrak{g}$ be a Kac-Moody algebra, $W$ be its Weyl group, generate by fundmental reflections {$r_1,...,r_n$}. For $i,j$, we have relation $(r_ir_j)^{m_{ij}}=1$, $1\le i,j\le n$. Let $W^{'}$ be ...
fool rabbit's user avatar
2 votes
0 answers
179 views

Positive roots and the longest element of the Weyl group

Take $\frak{g}$ a complex semisimple Lie algebra and its Weyl group $W$. Is it true that the number of positive roots of $\frak{g}$ is equal to the length of the longest element of $W$?
Boris Henriques's user avatar
1 vote
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129 views

Relation between C-groups and reflection groups

Take a set of reflections $\{r_1,\ldots,r_k\}$ of $\mathbb R^n$. Sometimes, the group presentation will turn out to be a C-group – this is where the regular planar polytopes in Euclidean space, ...
ViHdzP's user avatar
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10 votes
1 answer
179 views

Reversals of autonomous subsets in right-angled Coxeter groups

This question has to do with some experimental phenomenon in groups generated by involutions that I cannot explain. Let $G$ be a finite, undirected graph, and let $W$ be the corresponding right-angled ...
Sam Hopkins's user avatar
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0 votes
0 answers
83 views

Reference request: Weyl group action on the power set of positive roots

There is a symmetric group action on the power set of positive roots in type A. The action is defined as follows. Denote by $\alpha_1, \ldots, \alpha_n$ be the set of simple roots in a root system. In ...
Jianrong Li's user avatar
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3 votes
1 answer
65 views

Action of Coxeter element on mod $2$ root lattice is semisimple

Let $\Lambda$ be a simply laced root lattice and $w$ a Coxeter element of the Weyl group of $\Lambda$. Question: Is it true that the action of $w$ on the $\mathbb{F}_2$-vector space $\Lambda/2\Lambda$ ...
Jef's user avatar
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3 votes
0 answers
79 views

Algebraic hypersurfaces and Coxeter groups

What is the minimum degree of an algebraic hypersurface (not necessarily smooth) having each Coxeter group as its symmetry group?
Daniel Sebald's user avatar
0 votes
0 answers
148 views

Coxeter groups and finitness of number of roots

Take any graph $\Gamma$ with $n$ vertices $\{v_1, v_2 \dots v_n\}$, and associate to this graph it's set of simple roots i.e. a vectors of the canonical basis $e_i, \ i=1..n$ of $R^n$ for each vertex $...
Gianfranco's user avatar
3 votes
1 answer
254 views

Infinite reflection subgroups of affine Coxeter groups

Let $(W,S)$ be an irreducible affine Coxeter system of rank $n \geq 3$ (affine for instance as in the sense of Chapter 4 of Humphreys "Reflection groups and Coxeter groups"). Let $t_1,\ldots,...
P. Wegener's user avatar
2 votes
1 answer
61 views

Symmetric algebra over a realization of Coxeter System is a dgg algebra

I have been reading a paper of Achar, Makisumi, Riche and Williamson. In the chapter 3, the authors talk us of bigraded modules and dgg modules and I'm stuck here. Let $(W, S)$ be a Coxeter system, ...
Rovil's user avatar
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2 votes
0 answers
194 views

Geometric or combinatorial interpretations of the (weak) Bruhat order?

$\DeclareMathOperator\Inv{Inv}$The weak Bruhat order on the symmetric group has a straightforward combinatorial interpretation: Consider a set of labelled balls $1,2,\dotsc,n$. Then for two ...
Brendan Mallery's user avatar
0 votes
1 answer
190 views

coset poset of reflection subgroup

Fix a finitely generated Coxeter system $(W, S)$, and let $W_J$ denote the standard parabolic proper subgroup generated by a subset $J \subset S$. It is well known that the poset of cosets $\{xW_J\}$ ...
J.D.Chern's user avatar
0 votes
1 answer
341 views

Are all Coxeter groups virtually free or virtually surface groups?

From Surface subgroups of Coxeter and Artin groups (Gordon, Long and Reid, 2003) DOI link, we can read that (Theorem 1.1) a Coxeter group is either virtually free or contains a surface group ($\pi_1$ ...
Jacques's user avatar
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3 votes
0 answers
168 views

The group of fixed points of an involution of a Weyl group

Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$. Let $W=W(R)$ denote its Weyl group. Let $S\subset R$ be a basis of $R$ (a system of simple roots). Let $D=D(R,S)$ denote the ...
Mikhail Borovoi's user avatar
8 votes
3 answers
415 views

Does $O(4,\mathbb{Q})$ have an exceptional outer automorphism?

Does the orthogonal group $O(4,\mathbb{Q})$ have an exceptional outer automorphism analogous to that of its subgroup, the Coxeter/Weyl group $W(F_4)$?
Daniel Sebald's user avatar
9 votes
3 answers
539 views

Subgroups of RAAGs vs. subgroups of RACGs

Is a (finitely generated) torsion-free subgroup of a right-angled Coxeter group isomorphic to a subgroup of a right-angled Artin group? It is well-known from the theory of special cube complexes that ...
AGenevois's user avatar
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3 votes
0 answers
116 views

What total orders have people studied on Coxeter Groups?

I'm aware of the ShortLex total order that gives rise to the usual normal form. But are there any others that have naturally arose and people have studied?
Rob Nicolaides's user avatar
2 votes
1 answer
202 views

When does a finite irreducible Coxeter Group act on the cosets of a parabolic subgroup faithfully?

Let $(W,S)$ be a finite and irreducible Coxeter Group. For $J \subseteq S$, let $W_J = \langle s | s \in J \rangle$, a parabolic subgroup. For which $J$ is the action (group multiplication on the left)...
Rob Nicolaides's user avatar
6 votes
1 answer
273 views

Swapping non-commuting generators in Coxeter group

Let $a$ and $b$ be two generators in a Coxeter group which do not commute. Is it possible for $ab$ to be equal to a product of generators where all instances of $b$ come before all instances of $a$? I'...
ViHdzP's user avatar
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