# 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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### 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 $\...

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### Can $E_8$ be enlarged?

Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?

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### When are indiscrete reflection groups Coxeter groups?

A well-known theorem of Coxeter states that any discrete group $W$ which is generated by reflections across (possibly affine) hyperplanes in Euclidean space is a Coxeter group: it has a presentation ...

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### Product of two reflections lying in a parabolic subgroup of a Coxeter group

Let $(W,S)$ be a Coxeter group, $I\subseteq S$ a subset of simple reflections, and $W_I \subseteq W$ the corresponding parabolic subgroup (we could also assume $|W_I|<\infty$, if needed).
Let also ...

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### Absolutely irreducible finite reflection/rotation groups

Suppose that $G$ is a finite irreducible reflection group with irreducible orthogonal representation $\rho: G\rightarrow \mathrm{O}(d)$, and let $\rho^+: G^+\rightarrow \mathrm{SO}(d)$ be its ...

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### Name for an involution associated to a Coxeter element

Let $(W,S)$ be a finite Coxeter system, and $c \in W$ a Coxeter element.
There is an involution $g\in W$ for which the involutive map $w \mapsto gw^{-1}g$ fixes $c$. Is there a standard name for this ...

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### Standard involutions conjugate to the negative of a standard involution in a Coxeter group

Consider a finite irreducible Coxeter group $W$ with a fixed generator set $S$. Every involution in $W$ is conjugate to a standard involution $c_I$, for some subset $I\subset S$. For example, this ...

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### Parabolic invariants of coinvariant algebras for reflection groups

Let $W$ be a finite reflection group and $V$ its reflection representation (over $\mathbb{C}$). Let $S$ be the symmetric algebra on $V^*$, $I_W\subseteq S$ the ideal generated by the non-constant $W$-...

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### Description of Soergel modules

So this is asking a basic and/or stupid question (my apology and appreciation) about Soergel modules that comes out of exercises by me who knows little about the subject.
Let $W$ be a finite Weyl ...

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### Decompositions of Coxeter groups into trees of groups

In Chapter 8.8 of Davis' "The geometry and topology of Coxeter groups" the smallest class $\mathcal{G}$ of Coxeter groups which contains all spherical Coxeter groups and which is closed ...

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### Algorithm for root system of Coxeter group generated by permutations

Suppose we are given a group $G$ in terms of generators $t_1, ..., t_n$ which are order 2 in $S_m$ (however we don't assume anything other than that these elements generate $G$ and have order 2). What ...

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### Motivation for the Kazhdan-Lusztig involution

I would like to know about the motivation behind the Kazhdan–Lusztig involution on an Iwahori–Hecke algebra.
I'll borrow the conventions from Libedinsky's Gentle introduction to Soergel bimodules I: ...

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### Why are fundamental weights denoted by omega?

In my field (and many others, I believe) the absolutely standard notation for the fundamental weights of a root system is lowercase omega: $\omega$. Recently I was taken aback to receive a copyedited ...

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### Words that give rise to an enumeration of elements of the symmetric group

Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...

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### Number of paths in the Bruhat order in the symmetric group

Let $\mathbb{S}_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}_m$, and consider paths in the Bruhat order like this: $1\lessdot v_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the ...

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### A graph similar to the Bruhat graph, what is it called?

The weak Bruhat graph (or 1-skeleton of the permutohedron) $B_n$ can be constructed as follows:
the vertices of $B_n$ are the permutations of the tuple $(1,...,n)$, two are joined by an edge, if they ...