Questions tagged [coxeter-groups]

For questions about Coxeter groups, an abstract group that admits a formal description in terms of reflections.

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120 cell generated from quaternions

The two quaternions $\omega={1\over 2}(-1,1,1,1)$ and $q={1\over 4}(0,2,\sqrt{5}+1,\sqrt{5}-1)$ generate a finite group under multiplication with 120 elements that form the vertices of a 600 cell, ...
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Number of inversions in $S_n$

I have a problem about some result appearing just before the proposition 1.5.2 in Combinatorics of Coxeter Groups by Bjorner, Brenti. It's about the inversions in $S_n$. I don't understand why 1.26) ...
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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 ...
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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 ...
• 7,082
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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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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 ...
• 7,082
1 vote
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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 ...
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Are reflection subgroups corresponding to closed root subsystems always parabolic?

In the third paragraph of this reference, the following is stated: let $W$ be a Coxeter group with set of roots $R$, and let $H$ be a subgroup of $W$ generated by reflections (i.e. by conjugates of ...
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Generating function for cardinality of Bruhat intervals

For a finite Coxeter system $(W, S)$, the Poincare polynomial counts the number of elements of a certain length: $$P_W(t) = \sum_{x \in W} t^{l(x)}.$$ The polynomials for types $A$, $B$, and $D$ are ...
• 11.5k
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Affine Permutations as a Coxeter Group

I was stuck trying to show that the group of $n$-affine permutations (i.e. bijections of $\mathbb{Z}$ such that $\sum_{i=1}^{n} f(i)-i =0$ and $f(i+n)=f(i)+n$ for a certain $n \in \mathbb{N}$) is a ...
• 53
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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 ...
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Conditions for finiteness of a reflection group

Given a subgroup $G$ of $O(V)$ that is generated by reflections ($V$ a Euclidean space), under what conditions can we determine that $G$ is finite? I realize that if the angles between any two roots ...
• 419
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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 ...
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Compute the lengths of Weyl group elements for all possible simple systems

So each choice of simple system for a root system determines a set of simple reflections. The length of an element of weyl group is the minimal number of simple reflections required to express it. My ...
• 419
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Hyperplane arrangements: Hyperplanes intersecting in a single point

I'm currently reading a text about hyperplane arrangements in a euclidian vector space V. The text says: [...] without loss of generality, we may assume that some of the hyperplanes intersect in a ...
• 2,126
1 vote
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Distance labels in regular hyperbolic tilings

Consider the order-4 pentagonal tiling of the hyperbolic plane (shown in the figure Hyperbolic plane tiling with pentagons). Pick a vertex $s$ (in white), label it with $0$ and then label all the ...
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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 $\mathcal{R}$ ...
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