# 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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### 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 ...

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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 ...

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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 ...

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### 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 $\{\...

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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 ...

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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 ...

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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 ...

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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 ...

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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?

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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} $...

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### 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 $w$ ...

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### 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 ...

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### 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 ...

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### 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)\...

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### 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 ...

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### 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 ...

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### 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 $H_s$...

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### 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 $B$...

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### 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 ...

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### 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 ...

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### 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 $W$?

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 $\Lambda/2\Lambda$ ...

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### 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?

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### 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 $...

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### 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,...

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### 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, ...

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### 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 ...

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### 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 $\{xW_J\}$ ...

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### 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 ($\pi_1$ ...

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### 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 ...

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### 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 $W(F_4)$?

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### 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 ...

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### 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?

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### 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)...

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### 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'...

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### Can Matsumoto's theorem for the symmetric group be proved using a monovariant?

This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions:
Let $n$ be a nonnegative integer. ...

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### Subgroups of $W(E_8)$

Are there any proper subgroups of the Coxeter group $W(E_8)$ which are also proper overgroups of $W(A_8)$, other than $\text{Aut}(A_8)$?

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### Order from Coxeter-Dynkin diagram

How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?

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### Indices of Coxeter groups in themselves

Every Euclidean Coxeter group ($P_n$, $Q_n$, $R_n$, $S_n$, $T_7$, $T_8$, $T_9$, $U_5$, $V_3$, $W_2$) contains infinitely many scaled copies of itself as subgroups. What are all the possible indices of ...

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### Algorithm to determine if a vector in the geometric representation of a Coxeter group is proportional to a root

Let $W$ be a Coxeter group, and let $V$ be its geometric representation (as defined for instance in Section 5.3 of Humphreys' book Reflection groups and Coxeter groups). Let $v\in V\backslash\{0\}$ (...

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### Isomorphic Coxeter groups

After enumerating the spherical Coxeter groups, it is easy to see that no two distinct cases are isomorphic. Does the same hold for Euclidean and hyperbolic Coxeter groups?

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### Which reflection groups can be enlarged?

Based on this question (which focuses on the case $E_8$) I wonder the following:
Question: For each finite reflection group $\Gamma\subseteq\mathrm O(\Bbb R^d)$, what is the largest finite group $\...