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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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.
Here are two examples of the kind of combinatorial abstractions of geometric ...
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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 ...
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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 ...
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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 ...
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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 "...
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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: ...
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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{...
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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 $...
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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” ...
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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 ...
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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 ...
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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 ...
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Recognizing reflection subgroups of Coxeter groups
Given a Coxeter system $(W,S)$ with reflections $T$, and any subset $A \subseteq T$, it is known that the reflection subgroup $W_A$ generated by $A$ has a canonical choice $S_A$ of generators so that $...
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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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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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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, ...
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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 ...
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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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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 ...
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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$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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(\...
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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 ...
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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 ...
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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.
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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,...
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$E_6$, $E_8$, 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 ...