# 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.

239 questions
Filter by
Sorted by
Tagged with
218 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 $w$ ...
• 467
223 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 ...
• 801
154 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 ...
• 31.4k
229 views

188 views

• 10.6k
451 views

### 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?
• 1,988
180 views

### 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 ...
• 2,553
530 views

### 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 ...
• 143
228 views

### 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 ...
• 439
274 views

### 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 ...
• 19.6k
1 vote
125 views

### 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 ...
55 views

### 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 $W$-...
• 2,850
191 views

### 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 ...
• 1,704
122 views

### 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 ...
163 views

### 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 ...
• 195
497 views

### 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: ...
262 views

### 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 ...
• 3,028
119 views

### 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 ...
• 952
207 views

### 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 ...
• 952
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 ...