All Questions
Tagged with group-theory coxeter-groups
127
questions
4
votes
2
answers
115
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Does every reflection generating set of a RA Coxeter group contain a conjugate of every standard generator?
I am interested in understanding generating sets of right-angle Coxeter groups (RACGs) consisting of reflections. More precisely, let $(W,S)$ be a finite rank RACG, and write $R=\{wsw^{-1}\mid s\in S\;...
- 2,382
3
votes
0
answers
55
views
A question regarding affine Coxeter groups
Let $\Gamma_n$ denote the isometry group of the regular tessellation of $\mathbb{R}^n$ by $n$-cubes, i.e. $\Gamma_n= \left( \bigoplus\limits_{i=1}^n \mathbb{D}_\infty \right) \rtimes S_n$. Now, let $...
- 33.3k
2
votes
0
answers
51
views
Generation function for the finite Coxeter group of type $D_k$
Basically I'd like to know how to derive the generation function for the finite Coxeter group of type $D_k$ to be familiar with notes in OEIS A162288:
According to formula section:
'The growth series ...
3
votes
1
answer
84
views
The Bruhat Orders of (finite irreducible) Coxeter Groups as Polytopes
The Strong Bruhat Order of a finite irreducible Coxeter Group satisfies all the axioms of being an abstract polytope. It's also a remarkably nice fact that the Weak Bruhat Order of the Symmetric Group ...
- 317
1
vote
1
answer
195
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Longest element of $D_n$ and the set of positive roots [duplicate]
Personally I am not very familiar with group theory and need some clarifications.
Let's look at $D_n$ and its longest elemements.
According to OEIS A162206 the triangle begins:
$1$;
$1;2;1$;
$1;3;5;6;...
2
votes
0
answers
69
views
Is there a neat way to construct the Coxeter Group $H_4$ from that of $H_3$ using the fact that $|H_4| = 14400 = 120^2 =|H_3|^2$?
Let $H_4$ and $H_3$ be the usual finite irreducible Coxeter Groups of their names: as presentations $H_4 = \langle s_1,s_2,s_3,s_4\rangle$ subject to the relations $$s_i^2 = (s_is_j)^2 = (s_1s_2)^5 = (...
- 317
3
votes
2
answers
208
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Defining a Coxeter group using all reflections
Let $(W, S)$ be a pair of a group $W$ and a subset $S$ consisting of involutions of $W$. We can consider the group $\tilde W$ with presentation $\langle S | \mathcal{R} \rangle$ where $\mathcal{R}$ ...
- 3,252
1
vote
0
answers
31
views
How can we define a reflection ordering without the use of roots?
In Björner and Brenti's 'Combinatorics of Coxeter Groups' (pg 137) there is the definition of a reflection ordering for a Coxeter Group: let $(W,S)$ be a Coxeter Group with some induced root system $\...
- 317
3
votes
1
answer
116
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Coxeter notation for the symmetries of the maximally symmetric unit-distance embedding of $K_{3,3}$ in $\mathbb R^4$
My Shibuya repository now contains unit-distance embeddings in the plane of all cubic symmetric graphs to $120$ vertices, except the first two ($K_4$ and $K_{3,3}$) which do not have this property, as ...
- 104k
1
vote
0
answers
141
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Which Coxeter Elements have powers that are the longest element of the (Finite, Irreducible) Coxeter Group?
Let $(W,S)$ be a finite, irreducible Coxeter Group. I thought it was true (from Humphrey's book Ex 2 on page 82) that if the Coxeter Number of $W$, $h$, is even then $$c^{h/2} = \omega_0 \quad \dagger$...
- 317
0
votes
0
answers
150
views
Is there a group with finitely many generators of order 2 that is not a Coxeter group?
Maybe I am missing something obvious here. But in this series of lectures the following is proved to be equivalent under the assumption that the group $W$ has a (finite) set of generators $S$ whose ...
- 475
2
votes
1
answer
178
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In the weak Bruhat order, is every element bounded by a power of a Coxeter element?
Let $(W,S)$ be a Coxeter System with $|W| = \infty$. Let $c = s_1\ldots s_{|S|}$ for some total ordering of $S$, a Coxeter element of $W$.
Is it true that for all $w \in W$, there exists a $k\in \...
- 317
0
votes
0
answers
70
views
When is the subgroup product of two parabolic subgroups of a Coxeter Group, the Coxeter Group itself?
Let $W$ be a Coxeter Group generated by simple reflections $S$. If $I,J\subseteq S$ and $W_I = \langle s | s \in I \rangle$ when is it true that $W_IW_J = W$?
I am secretly hoping that the answer ...
- 317
4
votes
0
answers
233
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How to find generators of translation subgroup of an abstract reflection (coxeter) group
I have an infinite reflection group https://en.wikipedia.org/wiki/Coxeter_group
Take for example the affine groups $[4,4],[4,3,4],[4,3,3,4]$...
I'd like to get an explicit expression for generators ...
- 1,010
2
votes
1
answer
97
views
Embedding of the 1-skeleton of a Coxeter group into its Davis complex
Let $(W,S)$ be a Coxeter system and let $\Sigma$ be the corresponding Davis complex. It is well-known that the Davis complex may be equipped with a piecewise Euclidean metric so that it is a proper, ...
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