Questions tagged [coxeter-groups]

A Coxeter group is a group defined by a presentation by involutions $r_i$ with relators $(r_ir_j)^{m_{ij}}=1$ for certain family $(m_{ij})$ of integers greater than 1.

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coset poset of reflection subgroup

Fix a finitely generated Coxeter system $(W, S)$, and let $W_J$ denote the standard parabolic proper subgroup generated by a subset $J \subset S$. It is well known that the poset of cosets $\{xW_J\}$ ...
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Does $O(4,\mathbb{Q})$ have an exceptional outer automorphism?

Does the orthogonal group $O(4,\mathbb{Q})$ have an exceptional outer automorphism analogous to that of its subgroup, the Coxeter/Weyl group $W(F_4)$?
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Are all Coxeter groups virtually free or virtually surface groups?

From Surface subgroups of Coxeter and Artin groups (Gordon, Long and Reid, 2003) DOI link, we can read that (Theorem 1.1) a Coxeter group is either virtually free or contains a surface group ($\pi_1$ ...
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The group of fixed points of an involution of a Weyl group

Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$. Let $W=W(R)$ denote its Weyl group. Let $S\subset R$ be a basis of $R$ (a system of simple roots). Let $D=D(R,S)$ denote the ...
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Coxeter subgroups of Coxeter groups

Is there an algorithm to determine all the Coxeter subgroups of a given Coxeter group? If we only want the Coxeter subgroups of finite index does that make the question easier? If we only want a ...
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Subgroups of RAAGs vs. subgroups of RACGs

Is a (finitely generated) torsion-free subgroup of a right-angled Coxeter group isomorphic to a subgroup of a right-angled Artin group? It is well-known from the theory of special cube complexes that ...
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When does a finite irreducible Coxeter Group act on the cosets of a parabolic subgroup faithfully?

Let $(W,S)$ be a finite and irreducible Coxeter Group. For $J \subseteq S$, let $W_J = \langle s | s \in J \rangle$, a parabolic subgroup. For which $J$ is the action (group multiplication on the left)...
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What total orders have people studied on Coxeter Groups?

I'm aware of the ShortLex total order that gives rise to the usual normal form. But are there any others that have naturally arose and people have studied?
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Completely positive maps on Coxeter groups - the general case

In the reference below Bozejko and Speicher showed the following (for the full statement see the remark on page 9): Let $(W,S)$ be a Coxeter system, let $\mathcal{H}$ be a Hilbert space and denote ...
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228 views

Swapping non-commuting generators in Coxeter group

Let $a$ and $b$ be two generators in a Coxeter group which do not commute. Is it possible for $ab$ to be equal to a product of generators where all instances of $b$ come before all instances of $a$? I'...
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Can Matsumoto's theorem for the symmetric group be proved using a monovariant?

This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions: Let $n$ be a nonnegative integer. ...
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Name for an involution associated to a Coxeter element

Let $(W,S)$ be a finite Coxeter system, and $c \in W$ a Coxeter element. There is an involution $g\in W$ for which the involutive map $w \mapsto gw^{-1}g$ fixes $c$. Is there a standard name for this ...
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Subgroups of $W(E_8)$

Are there any proper subgroups of the Coxeter group $W(E_8)$ which are also proper overgroups of $W(A_8)$, other than $\text{Aut}(A_8)$?
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Order from Coxeter-Dynkin diagram

How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?
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Indices of Coxeter groups in themselves

Every Euclidean Coxeter group ($P_n$, $Q_n$, $R_n$, $S_n$, $T_7$, $T_8$, $T_9$, $U_5$, $V_3$, $W_2$) contains infinitely many scaled copies of itself as subgroups. What are all the possible indices of ...
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Coxeter groups generated by one finite conjugacy class

Let $(W,S)$ be an arbitrary Coxeter system. We consider the following scenario: Let $\mathcal{O}$ be a conjugacy class of an element $w$ in $W$ which is finite and which generates the whole group $W$....
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Can $E_8$ be enlarged?

Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?
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Standard involutions conjugate to the negative of a standard involution in a Coxeter group

Consider a finite irreducible Coxeter group $W$ with a fixed generator set $S$. Every involution in $W$ is conjugate to a standard involution $c_I$, for some subset $I\subset S$. For example, this ...
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1answer
115 views

Algorithm to determine if a vector in the geometric representation of a Coxeter group is proportional to a root

Let $W$ be a Coxeter group, and let $V$ be its geometric representation (as defined for instance in Section 5.3 of Humphreys' book Reflection groups and Coxeter groups). Let $v\in V\backslash\{0\}$ (...
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Isomorphic Coxeter groups

After enumerating the spherical Coxeter groups, it is easy to see that no two distinct cases are isomorphic. Does the same hold for Euclidean and hyperbolic Coxeter groups?
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Which reflection groups can be enlarged?

Based on this question (which focuses on the case $E_8$) I wonder the following: Question: For each finite reflection group $\Gamma\subseteq\mathrm O(\Bbb R^d)$, what is the largest finite group $\...
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When are indiscrete reflection groups Coxeter groups?

A well-known theorem of Coxeter states that any discrete group $W$ which is generated by reflections across (possibly affine) hyperplanes in Euclidean space is a Coxeter group: it has a presentation ...
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Product of two reflections lying in a parabolic subgroup of a Coxeter group

Let $(W,S)$ be a Coxeter group, $I\subseteq S$ a subset of simple reflections, and $W_I \subseteq W$ the corresponding parabolic subgroup (we could also assume $|W_I|<\infty$, if needed). Let also ...
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483 views

Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
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Absolutely irreducible finite reflection/rotation groups

Suppose that $G$ is a finite irreducible reflection group with irreducible orthogonal representation $\rho: G\rightarrow \mathrm{O}(d)$, and let $\rho^+: G^+\rightarrow \mathrm{SO}(d)$ be its ...
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Motivation for the Kazhdan-Lusztig involution

I would like to know about the motivation behind the Kazhdan–Lusztig involution on an Iwahori–Hecke algebra. I'll borrow the conventions from Libedinsky's Gentle introduction to Soergel bimodules I: ...
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Parabolic invariants of coinvariant algebras for reflection groups

Let $W$ be a finite reflection group and $V$ its reflection representation (over $\mathbb{C}$). Let $S$ be the symmetric algebra on $V^*$, $I_W\subseteq S$ the ideal generated by the non-constant $W$-...
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1answer
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Description of Soergel modules

So this is asking a basic and/or stupid question (my apology and appreciation) about Soergel modules that comes out of exercises by me who knows little about the subject. Let $W$ be a finite Weyl ...
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1answer
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Decompositions of Coxeter groups into trees of groups

In Chapter 8.8 of Davis' "The geometry and topology of Coxeter groups" the smallest class $\mathcal{G}$ of Coxeter groups which contains all spherical Coxeter groups and which is closed ...
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Algorithm for root system of Coxeter group generated by permutations

Suppose we are given a group $G$ in terms of generators $t_1, ..., t_n$ which are order 2 in $S_m$ (however we don't assume anything other than that these elements generate $G$ and have order 2). What ...
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Why do combinatorial abstractions of geometric objects behave so well?

This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference: http://www.math.harvard.edu/cdm/. Here are two examples of the kind of ...
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Why are fundamental weights denoted by omega?

In my field (and many others, I believe) the absolutely standard notation for the fundamental weights of a root system is lowercase omega: $\omega$. Recently I was taken aback to receive a copyedited ...
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Words that give rise to an enumeration of elements of the symmetric group

Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...
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Number of paths in the Bruhat order in the symmetric group

Let $\mathbb{S}_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}_m$, and consider paths in the Bruhat order like this: $1\lessdot v_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the ...
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A graph similar to the Bruhat graph, what is it called?

The weak Bruhat graph (or 1-skeleton of the permutohedron) $B_n$ can be constructed as follows: the vertices of $B_n$ are the permutations of the tuple $(1,...,n)$, two are joined by an edge, if they ...
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1answer
256 views

Are orbit polytopes of rotation subgroup of Coxeter group combinatorially equivalent?

Suppose that $G\subset O(d)$ is a finite reflection (finite Coxeter) group. For any $v\in \mathbb{R}^d$ which is not fixed by any non-trivial $g\in G$, one can consider the orbit polytope (Coxeter) ...
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Are cyclic orbitopes of permutahedra necessarily simplicies?

Suppose that $v=(v_1,\ldots, v_d)\in \mathbb{R}^d$ lies in the linear subspace $v_1+\cdots +v_d=0$, and moreover that the coordinates are pairwise distinct. The permutahedron \begin{equation} P(\...
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Kazhdan-Lusztig basis elements appearing in product with distinguished involution

My apologies if the below is too malformed to make sense. Let $(W,S)$ be the affine Weyl group of a reductive group $G$, and let $\{C_w\}$ be the Kazhdan-Lusztig $C$-basis (an answer in terms of the $...
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Regarding $F_4$ and $G_2$ Lie algebras, do there exist $F_n$ or $G_n$ families of Lie algebras?

For example, $E_6$ exceptional Lie algebra is part of the $E_n$ series of Lie algebras (Kac-Moody algebras). Are $F_4$ or $G_2$ maybe also parts of some $F_n$ or $G_n$ series of Lie algebras or are ...
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Lattice structure in the root poset

Let $W$ be a Coxeter group with simple generators $s_1$, $s_2$, ..., $s_r$. Let $\Phi^+$ be the corresponding positive root system, with $\alpha_i$ the positive root corresponding to $s_i$. Bjorner ...
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Quotient of Coxeter Group II

My last question on the quotients of the group $$H := \langle a, b, c \ | \ a^2, b^2, c^2, (ab)^2, (ac)^3, (bc)^7, (abc)^{19} \rangle$$ couldn't be completely answered, because the finiteness of the ...
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Quotient of Coxeter group

Since the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ seems to have resisted attacks of some powerful programs, I will turn to a group that seems to be a bit easier to analyse....
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Is {6,3,7} an 'ultrahyperbolic' Coxeter group?

These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the {6,3,7} Coxeter group, by which I mean the one coming from the Coxeter diagram $$\circ-6-\circ-3-\...
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Irreducibility of Coxeter graphs as a function of generating sets

Given a Coxeter system $(W, S)$, we can form its Coxeter graph, and say that the system is irreducible if the graph is connected. Now, irreducibility is not solely a function of $W$; it depends also ...
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1answer
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Eigenvalues for elements of (infinite) Coxeter groups

My current research requires some knowledge on the eigenvectors of elements (of infinite order) of Coxeter groups view as reflections in their geometric representation. After some reading, my ...
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1answer
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The least common multiple of all degrees of a finite Coxeter group and indecomposable elements in the generalized cycle decomposition

This question is a follow-up of the previous question and especially the last comment therein. Let $(W,S)$ be a finite Coxeter system with reflections $T$. Let $\ell_T$ be the reflection length. ...
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1answer
222 views

Recurrence relation for number of reduced words of longest element in $S_n$

Is there any recurrence relation known for the number of reduced words of the longest element in $S_n$ (not commutation classes)? Edit: Sorry for unaccepting the answer, but I realized that I really ...
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Carter Payne homomorphisms and reduced expressions

Let $G$ be an algebraic group and $W$ denote the underlying affine Weyl group. I will label representations of the principal block of $G$ by their alcoves, which in turn I label by the corresponding ...
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Stability of infinite root systems with a long path in their Coxeter diagrams

Given a Cartan matrix associated to a Coxeter diagram, I can modify it by replacing one of the edges in the diagram with a long chain of vertices connected by simply laced edges; for example, this is ...
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1answer
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Isometry type of alcoves in affine Coxeter complexes

Let $W$ be an irreducible affine Coxeter group (say of type $\widetilde{X}_n$), and let $\Sigma$ be the associated Coxeter complex. Thus, $\Sigma$ is an $n$-dimensional Euclidean space tesselated by ...

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