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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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On the non-negativity of the coefficients of the Kazhdan-Lusztig polynomial of two kinds of linear algebraic groups

For $(x_1,...,x_{2n+1}) \in \mathbb C^{2n+1}$, let $q(x_1,...,x_{2n+1})=x^2_{n+1}+\sum_{i=1}^n x_ix_{2n+2-i}$. Let $SO(2n+1):=\{A\in M(2n+1,\mathbb C) | \det A=1 $ and $q(A\bar x)=q(\bar x), \...
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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 ...
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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 ...
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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 ...
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How to translate an element in a Coxeter group written as a matrix in Sage to reflections (a list)?

I am trying to use Sage to reduce a word to a reduced word. For example, consider the word $w=[4, 3, 2, 4, 3, 2, 1, 2, 4, 3, 2, 1, 3]=s_4s_3s_2s_4s_3s_2s_1s_2s_4s_3s_2s_1$. I used the following code ...
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Fundamental domains in Reflection groups and Coxeter groups - by Humphreys

In this thm I do not understand how he is using the induction in item d). The step t=1 is clear. enter image description here enter image description here
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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,...
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Can generators of a Coxeter group be redundant?

I’m just beginning to learn about Coxeter groups and at one point the textbook’s author seems to take it for grant that each generator cannot be “reduced” into a product of other generators. ...
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About parabolic subgroup of a Weyl group

Let $W$ be a Weyl group/Coxeter group. Let $\Phi$ be the associated root system, fix a positive root system $\Phi^+$ and let $\Delta$ be the set of simple roots. Let $W_I$ be the parabolic subgroup ...
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Reflection length of the longest word in a Coxeter group.

Let $(W,S)$ be a Coxeter system. The set of reflections of $W$ is $S=\{wsw^{-1}:s \in S, w \in W\}$. The reflection length of $w \in W$ is defined as the minimal number $m$ such that $w = r_1 \cdots ...
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How to show bilinearity

Given a Coxeter group $\varGamma =\langle \rho_0, \rho_1, \ldots, \rho_{n - 1}\rangle$ which at least satisfy the relation $(\rho_i\rho_j)^{p_{ij}}=1, \ \ 0\leq i, j \leq n - 1,$ where $p_{ii}=1$ and $...
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Type B Catalan numbers as signed permutations

The Catalan numbers are in bijection with the 123, 132, etc. avoiding permutations in $S_n$. If we move to type B, the type B Catalan numbers is $\binom{2n}{n}$, and the permutation group is the ...
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Coxeter length in the symmetric group equals number of inversions

Let $S_n$ be the symmetric group on the set $\{1,\dots, n\}$ and $\sigma\in S_n$. An inversion of $\sigma$ is a pair $(i,j)$ such that $1\leq i<j\leq n$ and $\sigma(i)>\sigma(j)$. Let $S_n$ be ...
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The ratio of Raw Maxima of Mahonian numbers in terms of groups

There is a clear explanation that the polynomial $$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$ with Mahonian numbers gives the Poincare series for the symmetric group $S_{n+1}$ considered as a Coxeter group ...
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Why $(1+x)(1+x+x^2)\cdots(1+x+x^2+\cdots+x^n)$ gives the Poincare series?

I am looking for an explanation of the fact that the polynomial $$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$ with Mahonian numbers gives the Poincare series for the symmetric group $S_{n+1}$ considered as a ...
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Restriction of Bruhat order to stabilizer of a vector

Let $W$ be a Weyl group (maybe better a Coxeter group, i.e. a group with action on vector space $V$ generated by reflections with some conditions). Consider $v \in V$ be a vector. Let $\text{Stab}_v \...
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Why does graphing a specific type of coxeter group result in a graph of an n-cube?

Specifically, graphing a coxeter group with coxeter matrix $M$ of size $n$ such that $M_{ii} = 1$ and $M_{ij} = 2$ in a graph where the vertices are elements of the group and an edge between $x$ and $...
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$B$-adapted pairs and $\hat{G}$ acting on $W$

Let $(G,B,N,S)$ be a Tits system, and let $\varphi: G \rightarrow \hat{G}$ be a $B$-adapted homomorphism. This means the kernel of $\varphi$ is contained in $G$, and for every $g \in \hat{G}$, there ...
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In a Bruhat-Tits building, $N$ is the stabilizer of the standard apartment

Let $(G,B,N,S)$ be a Tits system. Assume the system is saturated, which is to say that if we define $$H = \bigcap\limits_{n \in N} nBn^{-1}$$ then $H \subseteq N$. To an affine root system whose ...
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What is a Coxeter Group?

I've recently started investigating abstract algebra and have now stumbled upon "Coxeter Groups", which are a mystery to me. I've read that Coxeter Groups have something to do with reflections (in ...
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Kazhdan-Lusztig polynomials of a Dihedral group $D_n$

We want to compute the Kazhdan-Lusztig polynomials of a Dihedral group $D_n$. I understand that they are of the form $$(1,v,v,v^2,v^2,v^3,v^3,\cdots,v^k)$$ and am able to show it by hands in small ...
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Combinatorics of Coxeter Groups and lenght function

Let $W$ be a Coxeter Group with generators $s_1, \cdots, s_n$ and $l:W \to \mathbb N$ be the lenght function, and $T \subset W$ the subgroup of reflections. Then $$l(w)<l(tw) \iff l(w)<l(wt)$$ ...
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Two definitions of Bruhat order on $S_n$

Let $s_i = (i, i+1)$ be the $n-1$ transpositions that generate $S_n$. The (strong) Bruhat order says that $u \le v$ if for some reduced word $w$ composed of $s_i$ representing $v$, there is a (not ...
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Conjugating the longest element of a parabolic subgroup by longest element in larger parabolic?

This is a minor detail in a paper I'm reading. Let $W$ be a Weyl group with simple roots $\Pi$, and suppose $J\subseteq\Pi$. Let $w_I$ denote the longest element of the parabolic subgroup $W_I$ for $I\...
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How to obtain uniqueness in correspondence between simple systems and positive systems?

In reading the appendix of Lectures on Chevalley Groups by Steinberg, I'm having trouble understanding the uniqueness aspect of Proposition 9 (in both parts). Here is the setup. Let $V$ be an inner ...
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In what different terms can Coxeter systems be described?

My starting point is this question: https://mathoverflow.net/questions/214569 As I understand it they say, that the Coxeter matrix is not sufficient to describe the group. I thought that up to ...
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The coefficient of a root in a root system must be 0 or at least 1

Let W be a Coxeter group (not necessarily finite), and let Π and Φ be the corresponding root basis and root system. Suppose that x ∈ Φ + and a ∈ Π such that the coefficient of a in x is not zero. ...
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Cardinality of a coxeter group

Let ${G}$ be a Coxeter group with the next presentation \begin{equation} G = \left\langle s_1,s_2,\cdots,s_{n-1} : (s_is_{i+1})^3=1 , \ (s_is_j)^2=1 \ ,\ |i-j| > 1 \right\rangle \end{equation} ...
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Conditions for a neat subgroup to act fixed-point free

Given a hyperbolic reflection group $G$ acting on hyperbolic space $\mathbb{H}_n$ by, well, reflections in hyperplanes. Does a neat subgroup of $G$ act fixed-point free on $\mathbb{H}_n$? If not, ...
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Longest element of a subgroup

Say I have a finite Weyl group, $W$, and a set of generators $S:= \{s_1,...,s_k\}$ (making $W,S$ a coxeter system) and an automorphism $\theta: W\rightarrow W$ which permutes $S$. I know that the ...
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Is this “co-location” of two $E_6$'s, two $F_4$'s, and one $E_8$ possible?

Edited 1/5/2018 4pm US EDST: At the bottom of this post, I have added an email to Dr. Klizting - I sent this email to him after he posted his answer to the question. Question: Imagine three sets of ...
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Relating height to length in a root system

I have a question about a proposition from Casselman's notes on representation theory. Let $\mathfrak a$ be a finite dimensional vector space, and $\Sigma$ a reduced root system in the dual $\...
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If $E_6$ arises WITHIN $E_8$, does $E_8$ also arise FROM $E_6$?

Background: The standard treatments of the root-systems of $E_6$ and $E_8$ always speak of the roots of $E_6$ as being a subset of the roots of $E_8$, thereby implying that if we construct $E_8$ ...
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Finding orbits of fundamental weights

Let $\Phi$ be a classical root system and let $\omega_i $ be the i-th fundamental weight. I need to find the orbit of $\omega_i$ by the action of the Weyl Group $W$. How to compute it? I'm looking ...
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Is the 9-space coordinatizion of the roots of $E_6$ “nicely” related to the 8-space coordinatization of these roots as 72 roots of $E_8$?

Context for question: I am asking this question because my team can now: i) CERTAINLY show that biomolecular codon space (the space of "genes") instantiates two opposed instances of 4$_2$$_1$, ...
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Presentations of groups

I'm trying to understand presentations of groups better than I currently do. Does anyone know of any good online sources that are worth a look?
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A wall is a union of panels

Let $(W,S)$ be a Coxeter system. Let $A$ be the set of cosets $wW_{S-\{s\}} : w \in W, s \in S$. And for each $w \in W$, let $$C_w = \{ wW_{S-\{s\}} : s \in S \}$$ Then $A$ is an apartment with ...
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Reduced decomposition of the long element of $C_n$ or $B_n$?

Let $W$ be the Weyl group of the root system of type $C_n$. Then $W$ can be identified with the group of signed permutations on $1, 2, ... , n$. Let $S = \{s_1, ... , s_n\}$, where $s_i$ swaps $i$ ...
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Question about proof of positive roots under reflection

Let $(W, S)$ be a finite Coxeter system. Furthermore, let $V$ be a real vector space with a (finite) basis $\{ \alpha_s | s \in S \}$. For every $s \in S$ one can define the reflection $\sigma_s : V ...
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Parabolic Bruhat order

Let us consider $GL(V)$ over complex numbers. Let $P \subset G$ be a parabolic subgroup. Let $T \subset P$ be a maximal subtorus. Torus $T$ gives us a decomposition of $V$ into the sum of one-...
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Exercise 3 in Chapter IV, Section 1 on Bourbaki, *Lie Groups and Lie Algebras*

This is exercise 3 in Bourbaki, Lie Groups and Lie Algebras. The application I'm interested in is the fact that for any $\theta \subseteq \Delta$ in a reduced root system, there exists a unique left ...
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Exercise 1 in Chapter 4, Section 1 of Bourbaki, *Lie Groups and Lie Algebras*

This is the first exercise in Bourbaki, Lie Groups and Lie Algebras, Chapter IV.
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Exercise on saturated Tits system

I'm trying to work through the following exercise in Bourbaki, Lie Groups and Lie Algebras. For part (a), we are assuming $b \in B$ normalizes $N$, and we are supposed to show that $bnb^{-1}n^{-1} \...
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What does it mean for a Coxeter system to be of “spherical” type?

In the theorem of the paper Sur les valeurs propres de la transformation de Coxeter the author uses in the main theorem the term "spherical" to refer to a property that Coxeter systems $(W,S)$ can ...
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How to justify that any set of coxeter generators for a Weyl group are simple

Let $G$ be a group with a $\mathrm{BN}$-pair, and let $W:=N/(B\cap N)$ be the Weyl group of $G$, where $W$ is generated by a set $S$ of simple roots (as in the definition of a $\mathrm{BN}$-pair) and $...
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Reference request: Inversions and $\sum_{w\in W}q^{\ell(w)}$ for arbitrary Coxeter groups

For the symmetric group $S_n$, an inversion of a permutation $\pi∈S_n$ is a pair $1\leq i<j\leq n$ such that $\pi(i)>\pi(j)$. It is known that the length $\ell(\pi)$ of a permutation (i.e. the ...
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Alcoves in extended affine Weyl group of $A_1^{(1,1)}$

For affine Coxeter/Weyl groups, there is a standard representation into $GL_n(\mathbb{R})$ (where $n$ is the rank of the Coxeter group) where the reflecting hyperplanes form a nice hyperplane ...
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Conjugacy classes of Weyl group of type B

I'm really stuck in proving the fact that the conjugacy classes of B_n are deterimined by a positive and negative cycle stractures. Can you give me any hint. I think if I can use the fact that $g(i_1,...
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Orbits in Coxeter Group

I'm interested in computing the orbits in a finite coxeter group, i.e. $D_4$ (see here ). The orbits of the roots $\mathcal{O}(r_i)$ can be obtained by applying all the reflections $\displaystyle ...
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Confused by reflection formula for Coxeter systems of type $I_2(4)$ and $I_2(6)$ in Humphreys.

In Section 5.3 of Humphreys book on Coxeter/Reflection groups, he develops a geometric representation of a Coxeter system $(W,S)$ by taking an $\mathbb{R}$-vector spaces $V$ with basis $\{\alpha_s:s\...