All Questions
Tagged with group-theory coxeter-groups
127
questions
2
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answer
258
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Crystallographic root system Coxeter Groups
In Humphreys book in "Reflection Groups and Coxeter Groups" he defines a root system $\Phi$ to be crystallographic if it satisfies $\frac{2(\alpha, \beta)}{(\beta, \beta)} \in \mathbb{Z}$ $(\star)$ ...
- 121
3
votes
1
answer
57
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Partial Order on a Vector space with a Crystallographic Coxeter Group
Let $V$ be a vector space, and $W$ an irreducible, crystallographic Coxeter group on $V$ with simple root system $\Pi = \{a_1 \cdots a_n \}$.
We define a partial ordering on $V$ by $u \geq v$ iff $u - ...
- 127
3
votes
1
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88
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Orbits of pairs of multi-indices under the diagonal action of the symmetric group
This question concerns a statement made on page 168 of
Dipper, R. and Donkin, S., 1991. Quantum GLn. Proceedings of the London Mathematical Society, 3(1), pp.165-211.
I have tried to include all ...
- 2,251
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112
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Question about a certain involution on a Coxeter group $W$.
This is a small question that arose while reading the paper On Okuyama's Theorems About Alvis-Curtis Duality by M. Cabanes. It can be read here.
Let $(W,S)$ be a finite Coxeter system, $l$ the length ...
- 12.4k
1
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2
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190
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What are the word hyperbolic affine Coxeter groups?
It is well-knwon that all affine (irreducible) Coxeter systems can be classified by their Coxeter graphs, see Wikipedia. The corresponding diagrams are $(\tilde{A}_n)_{n \geq 1}$, $(\tilde{B}_n)_{n \...
- 937
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242
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Let $w_0$ be the element of longest length in a Coxeter group. Show that $l(w_0w)=l(ww_0)=l(w_0)-l(w)$? Find $w_0$ explicitly in $S_n$.
Let $w_0$ be the unique longest element in $W=S_n$. Let us show that $$l(ww_0)=l(w_0)-l(w)$$ for any $w \in W$.
We proceed by induction on $l(w)$. First, let $S$ be the generating set for $W$. In ...
- 229
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1
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There is an element of every possible length in $[W_{\theta} \backslash W]$
Let $(W,S)$ be the Weyl group of a root system with base $\Delta$, and let $\theta \subset \Delta$. Let $W_{\theta}$ be the group generated by $\theta$. It is a general result that in every right ...
- 34.3k
0
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63
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Coxeter graph of the group $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$
I am reading the first chapter of Combinatorics of Coxeter Groups by A.Björner and F.Brenti. In the first example they say that the graph with $n$ isolated vertices (no edges) is the Coxeter graph of ...
- 621
10
votes
3
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905
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What sort of groups are generated by a single conjugacy class?
To clarify, I am not looking for a classification but rather for well-researched examples of families of (finitely generated) groups generated by a single conjugacy class.
A collection of examples, ...
- 4,786
1
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0
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100
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Coxeter exchange condition in symmetric group
I would like to prove (for purposes of illustration mainly) that the symmetric group $S_n$ with the set $S$ of adjacent transpositions $(i, i+1)$ is a Coxeter group by proving that it satisfies the ...
- 103
2
votes
1
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274
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Motivation for definition of bilinear form in linear representation of Coxeter groups?
In a set of notes on Coxeter groups I am reading the following definitions are made:
Let $M = (m_{ij})_{1 \leq i,j \leq n}$ be a symmetric $n \times n$ matrix with entries from $\mathbb{N} \cup \...
- 2,501
3
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79
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How would one classify point groups?
By point group I mean a finit subgroup of $\mathrm O(\Bbb R^n)$.
Lists of point groups for some small dimensions are found on Wikipedia, but I am not certain about their completeness. As there seem ...
- 30.1k
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70
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Counting inversions of random elements in coxeter groups
I am trying to find a general interperetation to the following facts (pls be patient to read it).
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of ...
2
votes
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Embeddings of Coxeter Groups of Rank $3$ into $\text{SO}(3)$
Let $W = \langle x_1, x_2, x_3 \;|\; (x_ix_j)^{m_{ij}} \rangle$ be an irreducible Coxeter group, i.e., the graph with vertices $v_1, v_2, v_3$ and edges between all pairs $(v_i, v_j)$ with $m_{ij} \ne ...
- 4,786
1
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1
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93
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Can every positive root of a Coxeter group be written as a simple root and a positive root?
Can every positive root of a Coxeter group be written as a simple root and a positive root? I think that this is possible. For example, in type $B_2$, the set of positive roots are $\alpha_1, \alpha_2,...
- 14.6k