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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7
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0answers
118 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 ...
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1answer
150 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 ...
2
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1answer
203 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 ...
2
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30 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 ...
4
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51 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 ...
2
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1answer
162 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 ...
6
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1answer
119 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 ...
4
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1answer
105 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 ...
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119 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. ...
9
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171 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 $...
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1answer
107 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 ...
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2answers
262 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 ...
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1answer
97 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. ...
3
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1answer
133 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', ...
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203 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 ...
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87 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)-\...
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131 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 ...
2
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124 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\...
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83 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_{...
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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 ...
2
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1answer
82 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 ...
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1answer
67 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 ...
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147 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 ...
6
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1answer
158 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 ...
2
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1answer
175 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 $x\le w$?
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107 views

Intersection of Levi subgroups via intersection of their Weyl groups

Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a maximal torus $T \subset G$. Let $M,L \subset G$ be its Levi subgroups containing $T$ (note that we do note assume that $M,L$ are ...
4
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1answer
174 views

Reduced expressions for reflections in a Coxeter group

Let $(W,S)$ be a Coxeter system and $\beta$ a positive root in it. Is there a good way to compute a reduced expression for the reflection across the hyperplane with normal $\beta$? References please.
2
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2answers
151 views

Greatest element of ${}^IW$

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $W$ be the associated Weyl group and let $\Phi$ be its root system. We write $\Phi^+...
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150 views

The growth of maximum elements for the reflection group $D_n$

Let's consider maximum coefficients in Poincaré polynomial (or growth series) for the reflection group (or Weyl group, or finite Coxeter group) $D_n$ as mentioned in A162206. The maximal numbers $M(n)$...
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17 views

Alcove address characterization of weak order reference request

Let $\Phi$ be the root system of type $A$. Let $\mathcal{A}$ be an alcove of the corresponding affine arrangement. The address (or Shi coordinates) of $\mathcal{A}$ is a function $k:\Phi^+ \rightarrow ...
6
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1answer
204 views

Vanishing of certain coefficients coming from Coxeter groups

Let $\left(W\text{, }S\right)$ be a Coxeter system. For every $w\in W$ let us write $\left|w\right|$ for the length of $w$. Set $\lambda\left(e\right)=1$ where $e\in W$ denotes the neutral element of ...
3
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48 views

Directed galleries of the building of type $\widetilde{A}_{n}$

Let $X$ be the affine building of $GL_n(\mathbb{Q}_{p})$. We call oreinetd chamber of $X$ every sequence $\overrightarrow{C}=(s_1,...,s_n)$ of vertices such that $C=\{s_1,...,s_n\}$ is a chamber of $X$...
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1answer
68 views

In Type $A$, if the Bruhat graph of an element $w$ in the Weyl group is regular, then to show that $l(w)=$ # $ \{\alpha \in R^+| s_{\alpha} \le w\}$

I am trying to prove that for type $A$ , rational smoothness of Schubert varieties implies smoothness. So suppose we are in Type $A_{n-1}$, so let $G=Sl(n,\mathbb C)$, $B=$ the group of upper ...
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140 views

A certain kind of permutations and transport of Bruhat chains under conjugation

Let $(W,S)$ be a finite Coxeter system. Let us consider the following situation: Let $v_1,v_2,w\in W$ such that $v_1=wv_2w^{-1}$. Let $s_{\beta_r}\ldots s_{\beta_1}$ be a reduced expression of $v_2$. ...
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159 views

Relation between groups $A_n$, $B_n$, $D_n$ and $S_n$ or inversions of random elements in Coxeter groups

First of of all I'm trying to find a general interpretation to the following facts below. Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian ...
1
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1answer
129 views

The sign of a signed permutation

There are two notions of 'sign' for signed permutations: the parity of the length (that is, the minimal length of a reduced word) the parity of the number of signs in the one line notation. I will ...
4
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1answer
181 views

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 $...
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311 views

What is the centralizer of a Coxeter element?

Let $(W,S)$ be a Coxeter system (of finite rank) and $c \in W$ a Coxeter element. If $W$ is finite, then the centralizer $C_W(c)$ is the cyclic group generated by $c$ (e.g. see the book "Reflection ...
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4answers
179 views

Upper and lower bounds on the number of certain subsets of the power set

Let $A$ be a set with $n$ elements. Call a subset $C$ of the power set of $A$ "good" if Each element of $C$ has at least three elements. If $P, Q\in C$ and $P\cap Q$ has more than one element, then $...
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0answers
113 views

Word length norm in the symmetric group $\mathfrak{S}_r$

Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...
6
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0answers
348 views

Symmetry group and irreducible representation

Let $S$ be a bounded geometric shape in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(S)$ of $S$, and assume that $G(S) ...
3
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0answers
118 views

Reference request: which elements in a Coxeter group has longest reflection length?

Let $V$ be a vector space over $\mathbb{R}$. An element $s \in GL(V)$ is a reflection if $H_s:=\ker(s-1)$ is a hyperplane and $s^2=1$. The eigenvalues of a reflection $s$ are $1, -1$. Every reflection ...
0
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0answers
86 views

Coxeter group action on the product of root systems

Let W be a finite Coxeter group and $\Phi^+$ the set of its positive roots. The Coxeter group acts on $\Phi^+$ by $(w, \alpha) \mapsto w \cdot \alpha$ if $w \cdot \alpha \in \Phi^+$ and $(w, \alpha) \...
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47 views

Question about Ext group in $\mathcal{O}^\mathfrak{p}$?

Let $W$ be a Weyl group, let $\mu$ be an antidominant weight. Let $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$. Denote ${}^IW$ the set of minimal length coset ...
1
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0answers
67 views

About Kazhdan Lusztig polynomial evaluating at q=1

Given $w\le w'$ (in Bruhat ordering), does $P_{x,w}(1)\le P_{x,w'}(1)$ (in usual ordering of $\mathbb{R}$), where $P_{x,w}(q)$ is the Kazhdan Lusztig polynomial?
7
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1answer
118 views

How many maximal length Bruhat paths from $u$ to $w$ can there be?

I've been doing some work with saturated Bruhat paths in a Coxeter group between two elements $u\leq w$. It seems to me that if $\ell(u) =0$, then there are at most $\ell(w)! $. I haven't tried to ...
9
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2answers
236 views

Reference Request: Length of a reflection in a Coxeter group can be achieved by symmetric word

In a given coxeter group $(W,S)$, a reflection is an element of $W$ that can be written with a symmetric word in the generators $S$. In multiple sources, I found the following formula: $$ \mathrm{...
0
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1answer
150 views

About generator of minimal length coset representatives

Let $(W, S)$ be a Coxeter system. Let $J \subseteq S$ and recall that $W_J = \langle s: s \in J\rangle \subseteq W$. Define $W^J = \{w \in W : \ell(ws) > \ell(w)\ \text{for all } s \in J\}$. ...
1
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1answer
99 views

Closed subsets in Coxeter groups

Let $W$ be a finite or infinite Coxeter group and $\Phi^+$ the set of its positive roots. In the paper, a subset $A$ of $\Phi^+$ is closed if for all $a, b \in A$, $r_1 a + r_2 b \in \Phi^+$ for ...
1
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1answer
178 views

Reference request: Catalan number of type B

Are there some generalized Catalan number of type $B$ such that the sequence of the numbers is $3,9,29,97,333$ for $n=2,3,4,5,6$? As discussed in this previous question, there are at least two types ...