All Questions
Tagged with group-theory coxeter-groups
127
questions
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Coxeter Groups Classification Proof
I have recently noticed that Bourbaki's proof (Chapter 6, section 4, no.1, Lemma 10, on p205) of the classification of finite Coxeter groups uses the inequality
$$ \frac{1}{1+p} + \frac{1}{1+q} + \...
- 168
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65
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Is there an algorithm to check that a subgroup of a CAT$(0)$ group is *not* quasiconvex?
Let $G$ be a finitely generated CAT$(0)$ group and $H$ a subgroup. If $H$ is quasiconvex then it is finitely generated, so we can immediately conclude that any non-finitely generated subgroup of $G$ ...
- 2,382
3
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72
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Davis Regular Tessellations of Spheres and Straight Line Coxeter Groups
In Davis' "Geometry and Topology of Coxeter Groups", section B.3, in particular Theorem B.3.1, there is a proof that every finite "straight line" Coxeter group is associated to a ...
- 168
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55
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Why a retraction from a building to an apartment is not isometric?
Let $X$ be an affine building and $\mathcal{A}$ a system of apartment. For any apartment $A\in \mathcal{A}$ and a chamber $C$ in $A$, let us consider the retraction $\rho=\rho_{A,C}\colon X\...
- 161
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63
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When does multiplying by an involution increase the Bruhat Order in the Symmetric Group?
Let $w \in \mathrm{Sym}(n)$ for some positive integer $n$. Let $r$ be an involution in $\mathrm{Sym}(n)$, and write it as the product of disjoint transpositions like so: $$r = \prod_{i=1}^k (a_i,b_i) $...
- 317
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65
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Angles of the Fundamental Alcove (Chamber?)
I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I ...
- 8,040
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175
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Reflection Group of Type $D_n$
Here is the description of the reflection group of type $D_n$ in Humphreys' book Reflection Groups and Coxeter Groups:
($D_n$, $n \ge 4)$ Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of ...
- 8,040
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1
answer
64
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Does the Coxeter group $C(D_n)$ have any "proper reflection quotients" except $C(A_{n-1})$?
Here, a reflection quotient is a surjective homomorphism between Coxeter groups mapping reflections to reflections. A reflection quotient $C(\Delta) \to C(\Gamma)$ is proper if it is not injective and ...
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1
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290
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Reflection Group of Type $C_n$
In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way:
Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of ...
- 8,040
1
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100
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Classification of Finite Coxeter Groups Bjorner
While Humphreys gives a classification of finite irreducible Coxeter groups by their "geometric representation", Bjorner and Brenti (Combinatorics of Coxeter Groups) leave it as an exercise ...
- 168
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168
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Simple Reflections on Simple Roots
I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a ...
- 8,040
2
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1
answer
133
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How to find the longest element in a double coset of a Weyl group using SageMath?
I asked a question about computing longest element in a double coset in AskSage
It has not been answered for a long time. So I asked it here.
Let $W$ be a finite Coxeter group. Denote by $W_I$ the ...
- 14.6k
1
vote
1
answer
232
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Defining the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in GAP
I am looking to define the Weyl group of type $D_n$ as a subgroup of Weyl group of type $B_n$ in the software GAP. In general, one can define these groups separately. For example, let's say
...
- 4,125
2
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22
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System of representatives in reflection groups and subgroups
I'm working on a paper from Steinberg (1974), "On a theorem of Pittie". The paper is mainly about roots and reflection groups. I'm having trouble understanding the proof of lemma 2.5(a) ...
- 21
2
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2
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53
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Generating sets of semi-direct products with $\mathbb{Z}_2$
Suppose a group $G$ splits as a semidirect product $N\rtimes\mathbb{Z}_2$, and let $\phi:G\to\mathbb{Z}_2$ the the associated quotient map. If I have a subset of elements $\{g_1,\dots,g_n,h\}$ of $G$ ...
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