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.

Filter by
Sorted by
Tagged with
6 votes
1 answer
503 views

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. ...
darij grinberg's user avatar
4 votes
1 answer
204 views

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 $\text{Aut}(A_8)$?
Daniel Sebald's user avatar
6 votes
3 answers
431 views

Order from Coxeter-Dynkin diagram

How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?
Daniel Sebald's user avatar
6 votes
0 answers
99 views

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 ...
Daniel Sebald's user avatar
2 votes
1 answer
139 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 $v\in V\backslash\{0\}$ (...
Thomas Gobet's user avatar
0 votes
0 answers
159 views

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?
Daniel Sebald's user avatar
8 votes
1 answer
345 views

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 $\...
M. Winter's user avatar
  • 12.5k
9 votes
1 answer
486 views

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?
Daniel Sebald's user avatar
7 votes
1 answer
219 views

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 ...
Christian Gaetz's user avatar
10 votes
2 answers
580 views

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 ...
Utf's user avatar
  • 165
3 votes
3 answers
308 views

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 ...
Bob's user avatar
  • 439
8 votes
0 answers
282 views

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 ...
Sam Hopkins's user avatar
  • 22.7k
1 vote
2 answers
167 views

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 ...
Communicative Algebra's user avatar
3 votes
0 answers
64 views

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 $W$-...
Alexander Woo's user avatar
3 votes
1 answer
239 views

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 ...
Cheng-Chiang Tsai's user avatar
3 votes
1 answer
145 views

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 ...
worldreporter's user avatar
4 votes
2 answers
206 views

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 ...
manzana's user avatar
  • 335
8 votes
3 answers
592 views

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: ...
Richard Hepworth's user avatar
7 votes
0 answers
332 views

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 ...
Igor Makhlin's user avatar
  • 3,493
4 votes
0 answers
141 views

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 ...
Christoph Mark's user avatar
4 votes
1 answer
274 views

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 ...
Christoph Mark's user avatar
5 votes
1 answer
250 views

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 ...
M. Winter's user avatar
  • 12.5k
2 votes
1 answer
285 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) ...
Bob's user avatar
  • 439
3 votes
1 answer
156 views

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(\...
Bob's user avatar
  • 439
5 votes
0 answers
191 views

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 $...
Stefan  Dawydiak's user avatar
7 votes
0 answers
174 views

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 ...
Domagoj Hranjec's user avatar
12 votes
2 answers
470 views

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 ...
David E Speyer's user avatar
2 votes
1 answer
246 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 ...
Bipolar Minds's user avatar
2 votes
0 answers
45 views

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 ...
Chris Bowman's user avatar
  • 1,191
4 votes
0 answers
64 views

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 ...
Will Dana's user avatar
  • 453
2 votes
1 answer
564 views

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 ...
user154087's user avatar
7 votes
1 answer
187 views

Is the number of commutation classes of reduced words of the longest element of $S_n$ even for $n\geq 3$?

Observably, the number of primitive sorting networks on $n$ elements (or the number of commutation classes of reduced words of the longest element of $S_n$) is even for $3\leq n\leq 15$. These are all ...
Matt Samuel's user avatar
  • 2,008
4 votes
1 answer
126 views

Copies of $\mathbb{Z}\oplus \mathbb{F}_2$ in non-affine, irreducible Coxeter groups

Let $\left(W,S\right)$ be a non-affine, irreducible Coxeter system and assume that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{Z}$ (this is equivalent to $W$ being not word hyperbolic). Does this ...
worldreporter's user avatar
8 votes
0 answers
194 views

Is the order complex of open Bruhat intervals polytopal?

Let $P$ be the Bruhat order of a Coxeter group, and let $s<t$ in $P$. The set $\Delta(s,t)$ of all chains of the open interval $(s,t)$ (called the order complex of $(s,t)$) is a simplicial complex. ...
Richard Stanley's user avatar
10 votes
0 answers
340 views

Recognizing reflection subgroups of Coxeter groups

Given a Coxeter system $(W,S)$ with reflections $T$, and any subset $A \subseteq T$, it is known that the reflection subgroup $W_A$ generated by $A$ has a canonical choice $S_A$ of generators so that $...
Christian Gaetz's user avatar
1 vote
1 answer
146 views

The Fano plane, stericated 6-simplex, and pentallated 6-simplex

According to this link: https://en.wikipedia.org/wiki/Stericated_6-simplexes the stericated 5-simplex "scal" has 105 vertices defined as permutations of (0,0,1,1,1,1,2). In the course of my team's ...
David Halitsky's user avatar
8 votes
2 answers
380 views

Rank matrices for type $D$ Bruhat order

Roughly, this question asks how the Bruhat (strong) order in type $D$ can be understood like the Bruhat orders in types A and B=C. I'll review how types A and B work before asking my question. As a ...
David E Speyer's user avatar
1 vote
1 answer
148 views

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. ...
Christoph Mark's user avatar
3 votes
1 answer
223 views

Reference request: Finite (multi-parameter) Iwahori-Hecke algebras are pairwise isomorphic

Let $(W,S)$ be a Coxeter system. Let $q=(q_s)_{s\in S} \in \mathbb{R}^{\text{#}S}$ be a tuple of positive real numbers with $q_s=q_t$ whenever $s$ and $t$ are conjugate to each other. Follwing Davis', ...
worldreporter's user avatar
4 votes
0 answers
226 views

Do these Zariski-dense subgroups of complex Chevalley group have non-empty intersection with this Bruhat cell?

Let $G$ be a complex Chevalley group (not necessarily adjoint type) with $\operatorname{\mathbb{C}-rank}\geq2$ and let $H$ be a normal subgroup of $G(\mathbb Z)$ with a finite index (so $H$ is Zariski ...
Ami's user avatar
  • 71
3 votes
0 answers
101 views

Kazhdan-Lusztig polynomials and the defect of a Bruhat interval

Let $(W,S)$ be a Coxeter system with length function $\ell$ and $T=\bigcup_{w\in W}wSw^{-1}$. Set $N(u,v):=\{t\in T: u< tu \le v\}$, $\overline{\ell}(u,v):=|N(u,v)|$, $\ell(u,v):=\ell(v)-\...
James Cheung's user avatar
  • 1,855
3 votes
0 answers
236 views

The maximal order of an element in a Coxeter group

Let $W$ be a finite Coxeter group. Let $$ N_W=\operatorname{max}_{g\in W}\operatorname{ord}(g) $$ where $\operatorname{ord}(g)$ denotes the order of an element $g$. By Fermat's little theorem, we ...
Christoph Mark's user avatar
2 votes
0 answers
154 views

Deodhar's inequality: when the equality holds?

Let $(W,S)$ be a Coxeter system, $T=\bigcup_{w\in W}wSw^{-1}$ and $\ell$ be the length function. It is well-known that one have the following Deodhar's inequality: Let $x\le y\le w$. Then $|\{r\...
James Cheung's user avatar
  • 1,855
2 votes
0 answers
113 views

Monotonicity Theorem of inverse Kazhdan Lusztig polynomials

Let $P_{x,w}$ and $Q_{x,w}$ be the Kazhdan Lusztig polynomial and the inverse Kazhdan Lusztig polynomial of Coxeter group $W$, respectively. i.e., $\sum_{x\le y\le z}(-1)^{\ell(y)-\ell(x)}P_{x,y}(q)Q_{...
James Cheung's user avatar
  • 1,855
5 votes
0 answers
113 views

Progress on the result about montonicity of Kazhdan Lustzig polynomials

I am reading the paper Masato Kobayashi---Combinatorics on Bruhat Graphs and Kazhdan-Lusztig Polynomials. Let $P_{x,w}$ be the Kazhdan Lusztig polynomial of $W$. There is a result about ...
James Cheung's user avatar
  • 1,855
2 votes
1 answer
133 views

Characterization of all $w$ in the Weyl group satisfying $w \geq w_l w_{l, \theta}$

Let $W$ be the Weyl group of a root system $\Phi$ with base $\Delta$ and system of positive roots $\Phi^+$. Let $S = \{ w_{\alpha} : \alpha \in \Delta \}$ be the set of simple reflections ...
D_S's user avatar
  • 6,100
2 votes
1 answer
152 views

Consequence of Lifting property of Bruhat ordering

I am reading the book: Anders Björner, Francesco Brenti --- Combinatorics of Coxeter Groups. I would like to know whether a variation of Corollary 2.2.8 is true. In other words, does the following ...
James Cheung's user avatar
  • 1,855
4 votes
0 answers
268 views

What are the normal subgroups of the finite Coxeter Groups of type Bn?

Let $B_n = \langle \rho_0,\rho_1,\ldots,\rho_{n-1} \rangle$ subject to the relations that $(\rho_i\rho_j)^{m_{i,j}} = id$ with $m_{i,i} = 1$, $m_{i,j} = 2$ for $|i-j|\ge 2$, $m_{i,i+1} =3$ for $0 \le ...
Rob Nicolaides's user avatar
6 votes
1 answer
187 views

Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups

Let $\left(W\text{, }S\right)$ be a Gromov hyperbolic Coxeter system and denote by $\partial W$ the corresponding Gromov boundary. For $z\in\partial W$ let $\alpha$, $\beta$ be infinite geodesic paths ...
worldreporter's user avatar
2 votes
1 answer
195 views

Reduced expression and Bruhat order

For $n\ge 3$. Let $s_1\cdots s_n$ be a reduced expression of $x$. Suppose $s_1\cdots s_{n-1}\le w$ and $s_2\cdots s_{n}\le w$. Does this imply $x\le w$?
James Cheung's user avatar
  • 1,855