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.

Filter by
Sorted by
Tagged with
9
votes
2answers
483 views

Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
19
votes
4answers
834 views

Are there Hamilton paths in Cayley graphs of Coxeter groups?

Hi everyone. I want to optimize certain computation on finite Coxeter groups $(W,S)$. Basically I compute the matrices $\rho(T_w)$ for all $w\in W$ of a matrix representation $H\to K^{d\times d}$ of ...
93
votes
3answers
4k views

Why do combinatorial abstractions of geometric objects behave so well?

This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference: http://www.math.harvard.edu/cdm/. Here are two examples of the kind of ...
34
votes
5answers
9k views

Definitions of Hecke algebras

There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are ...
33
votes
3answers
2k views

Is there any need to study Coxeter systems (W,S) with S infinite?

In their treatise Groupes et algebres de Lie, Bourbaki (no doubt heavily influenced by Tits) devoted Chapter IV (1968) to the general theory of what they dubbed "Coxeter systems" $(W,S)$ along with "...
12
votes
0answers
329 views

Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...
7
votes
1answer
367 views

Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE graphs?

Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ ...
14
votes
2answers
439 views

Coxeter exchanges in non-reduced words

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be). Define the greedy or Demazure product of $R$ as follows: ...
12
votes
2answers
498 views

Bounding weight multiplicities by number of certain Coxeter elements

This question concerns lower bounds of certain weight multiplicities in finite dimensional representations of algebraic groups (or Lie groups, Lie algebras). Let's say $G$ is a simple algebraic group ...
12
votes
3answers
1k views

A special tessellation

Let $P$ be a convex $n$-gon. Suppose that we have an infinite number of $P$s, and that each of them is colored either red or blue. Here, let us consider the following operations : Operation 1 : Place ...
12
votes
2answers
549 views

Generalization of cycle decomposition to Coxeter groups

I'm looking for a generalization of cycle decompositions for permutations to elements of Coxeter groups. (For the purposes of this question, any conjugate of a parabolic subgroup is also a parabolic ...
7
votes
1answer
382 views

Monotonic maximal chains in a Coxeter group

Let $(W, S)$ be a Coxeter system, and let $T = \bigcup_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha_t \in \Phi^{+}$ considered as a vector in ...
7
votes
4answers
848 views

Infinite Coxeter groups with a non-trivial finite conjugacy class?

Let $(W,S)$ be a Coxeter system, where $S$ is finite. Assume that $W$ has an infinite number of elements. Is it true that conjugacy classes of elements of non-central elements of $S$ have always an ...
4
votes
3answers
613 views

Maximal pairwise distance between $k$ permutations

How can k permutations on n-set be arranged to maximize minimal pairwise Kendall tau distance (i.e. number of discordant pairs) between them? For two permutations this is obviously when the second ...
14
votes
2answers
1k views

Random walks on Coxeter groups

Let $G_N$ be the group generated by elements $a_1,\ldots,a_N$ subject to the relations $a_i^2=1$ and $(a_ia_j)^3=1$. The growth function of $G_N$ is then $$f_N(t)=\frac{1+2t+2t^2+t^3}{1-Mt-Mt^2+\frac{...
8
votes
1answer
141 views

How many maximal length Bruhat paths from $u$ to $w$ can there be?

I've been doing some work with saturated Bruhat paths in a Coxeter group between two elements $u\leq w$. It seems to me that if $\ell(u) =0$, then there are at most $\ell(w)! $. I haven't tried to ...
7
votes
1answer
325 views

What is the relation between Coxeter transformations of generalized Cartan matrices and Coxeter transformations of finite-dimensional algebras?

Note: This question now has a sister :-) The Coxeter transformation of a generalized Cartan matrix: In the paper The spectral radius of the Coxeter transformations for a generalized Cartan matrix, ...
7
votes
0answers
151 views

Examples of reflection groups that are not Coxeter groups

Some background settings Let $V$ be a $n$-dimensinal vector space over $\mathbb{R}$, $V^\ast$ be its dual space. A reflection transformation $r$ on $V$ is a linear transformation that fixes a ...
1
vote
1answer
936 views

The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)

Original question (without additional information from Wendy): Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way: Taking the E8 as {128,...
1
vote
1answer
228 views

Type $C_n$ Weyl group contains in the centralizer of the longest word $w_0$ in $S_{2n}$

Are there some references about the proof of the following fact? Type $C_n$ Weyl group lies in the centralizer of the longest word $w_0$ in $S_{2n}$. Thank you very much.
14
votes
4answers
1k views

actions of the hyperoctahedral group

I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from $...
8
votes
1answer
311 views

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 $\...
8
votes
1answer
419 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?
6
votes
1answer
571 views

Coxeter subgroups of Coxeter groups

Is there an algorithm to determine all the Coxeter subgroups of a given Coxeter group? If we only want the Coxeter subgroups of finite index does that make the question easier? If we only want a ...
3
votes
1answer
138 views

Are cyclic orbitopes of permutahedra necessarily simplicies?

Suppose that $v=(v_1,\ldots, v_d)\in \mathbb{R}^d$ lies in the linear subspace $v_1+\cdots +v_d=0$, and moreover that the coordinates are pairwise distinct. The permutahedron \begin{equation} P(\...
3
votes
0answers
161 views

The maximal order of an element in a Coxeter group

Let $W$ be a finite Coxeter group. Let $$ N_W=\operatorname{max}_{g\in W}\operatorname{ord}(g) $$ where $\operatorname{ord}(g)$ denotes the order of an element $g$. By Fermat's little theorem, we ...
2
votes
1answer
171 views

Relation between Riemannian and Cayley-graph distance in a finite Coxeter group

Background: Let $W$ be a finite reflection group of rank $n$, acting on $\mathbb{R}^n$. The reflecting hyperplanes of $W$ meet the unit sphere $S^{n-1}\subset\mathbb{R}^n$, inducing a simplicial ...
1
vote
1answer
183 views

What is the relation between Coxeter transformations of Coxeter systems and Coxeter transformations of generalized Cartan matrices?

This question is related to the following question about Coxeter transformations that I asked and recently answered myself. For completeness I also write full definitions in the new question. The ...
-7
votes
4answers
912 views

$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes

Edited 1/21/2018 to add the following: Here is a DropBox link https://www.dropbox.com/s/7rtt0iqmgimsgzu/Zumkeller_edge-magic.pdf?dl=0 to a PDF showing how my team used biomolecular first ...