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Questions tagged [coxeter-groups]

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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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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 ...
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0answers
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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
90 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 ...
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2answers
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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{...
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1answer
56 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 $W^J = \{w \in W : \ell(ws) > \ell(w)\ \text{for all } s \in J\}$. ...
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1answer
92 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 ...
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1answer
133 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 ...
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Examples of reflection groups that are not Coxeter groups

Some background settings Let $V$ be a $n$-dimensinal vector space over $\mathbb{R}$, $V^\ast$ be its dual space. A reflection transformation $r$ on $V$ is a linear transformation that fixes a ...
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1answer
591 views

The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)

Original question (without additional information from Wendy): Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way: Taking the E8 as {128,...
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1answer
64 views

About Kazhdan-Lusztig polynomial

Let $(W,S)$ be a Coxeter system. One can have the Kazhdan-Lusztig polynomial $P_{x,\ y}(q)$. Does $P_{x,\ y}(q)=P_{x^{-1},\ y^{-1}}(q)$ for all $x,y\in W$?
5
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1answer
136 views

Parabolic Kazhdan-Lusztig polynomial coincide?

Let $(W,S)$ be a Coxeter system. For any subset $I\subseteq S$, we can have the parabolic Kazhdan-Lusztig polynomial $P_{x,w}^I(q)$ with respect to $I$. Now consider $I\subseteq J\subseteq S$. Both $(...
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0answers
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Ref. request: Enumerating elements of Bruhat cells

Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then $$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$ where we embed the ...
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1answer
143 views

Reference request: Coxeter length and irreducible characters

Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have $$\ell(\pi) = |\...
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An identity in Weyl group

Let $W$ be a Weyl group generated by the simple reflections $s_i$, $i \in I$, where $I$ is the vertex set of the Dynkin diagram of $W$. For $J \subset I$, let $W_J$ be the subgroup of $W$ generated by ...
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1answer
172 views

Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero

I am interested in how many pairs of permutations $(u,w)$ in $S_n$, such that the $\mu$-coefficient of its Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is non-zero? where $\mu_{u,w}=[q^{\frac{l(w)-l(u)-1}{...
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1answer
347 views

Coordinates of the Weyl vector of $E_8$ (and the 135 classes of $W(E_8)/W(D_8)$)

Consider the root system of $E_8$, written in its standard "even" coordinate system: i.e., it is the set of all $240$ vectors in $\mathbb{R}^8$ which whose coordinates are either all integers or all ...
3
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1answer
131 views

Elements of Coxeter group whose simple reflections pairwise commute

Let $W$ be a Coxeter group with associated graph $G$. Define $$X(G) = \{w \in W : \text{any two simple reflections} \,S_{\alpha}\, \text{and}\, S_{\beta} \,\text{appearing in any of the reduced ...
2
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1answer
106 views

Recursive formula for inverse Kazhdan-Lusztig polynomials

Let $(W,S)$ be a Coxeter group. Then the Kazhdan-Lusztig polynomials $P_{x,w}$ are known to satisfy the following recursive identity: For $x < w$ and $s \in S$ satisfying $\ell(sw) < \ell(w)$, ...
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1answer
73 views

Reduced decomposition for Weyl group elements which support a Bessel function

Let $\Delta$ be a set of simple roots for a reduced root system, and let $(W,S)$ be the associated Coxeter system, where $W$ is the Weyl group and $S$ is the set of simple reflections corresponding to ...
4
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1answer
136 views

Are descents in alternating subgroup counted by $h$-vector?

Consider the alternating subgroup $A_n$ of the symmetric group $S_n$ (or in general any Coxeter Group). Is there a simplicial complex whose $h$-vector $h_i$ equals the number of elements of $A_n$ with ...
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1answer
153 views

Dominance relation among Cartan matrices implies containment of root systems: Is this known?

Suppose $A$ and $A'$ are symmetrizable (generalized) Cartan matrices, in the sense of Kac's book Infinite-dimensional Lie algebras. Say $A$ dominates $A'$ if every entry of $A$ has weakly greater ...
5
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1answer
179 views

Instructions for using Coxeter 3.0 software

I am trying to use Coxeter 3.0 (http://www.liegroups.org/coxeter/coxeter3/english/coxeter3_e.html) to perform some computations for affine Weyl groups. I managed to install the program and get it ...
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3answers
762 views

$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes

Edited 1/21/2018 to add the following: Here is a DropBox link https://www.dropbox.com/s/7rtt0iqmgimsgzu/Zumkeller_edge-magic.pdf?dl=0 to a PDF showing how my team used biomolecular first ...
9
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1answer
197 views

A duality result for Coxeter groups

Short version: if $G$ is a Coxeter group and $H \subset G$ is a parabolic subgroup, both acting on a space $V$, is it true that the invariant-coinvariant algebra $(S(V)_G)^H$ has a natural bilinear ...
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0answers
286 views

A spin extension of a Coxeter group?

Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left( m_{s,t}\right) _{\left( s,t\right) \in S\times S}$. Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) ...
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1answer
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Question about proof of positive roots under reflection

Since I did not receive a lot of responses on Math Stack Exchange I would like to repost this question here. Let $(W, S)$ be a finite Coxeter system. Furthermore, let $V$ be a real vector space with ...
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2answers
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Have you seen my matroid?

Let $M(n,k)$ be the matroid on the ground set $\{\pm 1,\ldots,\pm n\}$ for which a set is independent if and only if it contains at most $k$ pairs $\pm i$. Note that the signed permutation group (the ...
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0answers
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Question on a reduction in Kirillov's paper on positivity of divided difference operators

As the title says, my question is on a specific argument in Kirillov - Skew divided difference operators and Schubert polynomials (journal, MSN) on positivity of divided difference operators. I recall ...
8
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1answer
250 views

Why is the root poset is graded by height?

Let $\Phi$ be a finite crytallographic root system. Let $\Phi^+$ be the positive roots and $\alpha_1$, ..., $\alpha_n$ be the simple roots. For $\beta = \sum c_i \alpha_i$ in $\Phi^+$, we define $h(\...
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1answer
131 views

Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE graphs?

Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ ...
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0answers
90 views

Type $C_n$ Weyl group contains in the centralizer of the longest word $w_0$ in $S_{2n}$

Are there some references about the proof of the following fact? Type $C_n$ Weyl group contains in the centralizer of the longest word $w_0$ in $S_{2n}$. Thank you very much.
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Characterization of permutations which have at most one successor in the covering relation of the weak Bruhat order

Let $W$ be the symmetric group on $n+1$ letters. Let $\ell$ be the length function on $W$. As the title says, can we characterize all $v\in W$ such that there exists a $w\in W$ such that for all ...
9
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1answer
197 views

Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$?

The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a_s,b_s)_{s \in S}$ one can ...
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2answers
169 views

A question about set of inversion

Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a ...
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1answer
97 views

What is the relation between Coxeter transformations of Coxeter systems and Coxeter transformations of generalized Cartan matrices?

This question is related to the following question about Coxeter transformations that I asked and recently answered myself. For completeness I also write full definitions in the new question. The ...
6
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1answer
187 views

What is the relation between Coxeter transformations of generalized Cartan matrices and Coxeter transformations of finite-dimensional algebras?

Note: This question now has a sister :-) The Coxeter transformation of a generalized Cartan matrix: In the paper The spectral radius of the Coxeter transformations for a generalized Cartan matrix, ...
7
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0answers
89 views

Number of occurrences of certain generators in expressions in Coxeter groups

Let $W$ be a Coxeter group (finite or infinite) with (finite) set $S$ of Coxeter generators, and let $I \subseteq S$ be some subset. If $w\in W$ then I call $m_I(w)$ the minimum total number of ...
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2answers
374 views

Bounding weight multiplicities by number of certain Coxeter elements

This question concerns lower bounds of certain weight multiplicities in finite dimensional representations of algebraic groups (or Lie groups, Lie algebras). Let's say $G$ is a simple algebraic group ...
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0answers
144 views

Reference request for generalized root systems

Where can I find information on root systems where the inner product is other than the standard (all positive) signature?
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0answers
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Have wiring diagrams been generalized to arbitrary digraphs?

A "combinatorial wiring diagram" is a way to define a permutation by a drawing of a particular planar digraph. For example, this wiring diagram corresponds to the permutation $(3412)$: In Coxeter ...
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1answer
174 views

Reference request: Reduced reflection length in Coxeter groups

I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...
1
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1answer
152 views

Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$

Looking at the images below, you recognize that the adjacency matrix of the graph $A_G$ splits up into three different colored submatrices, with $A_G=A_r+A_b+A_d$ (where $d$ is dark, damn...). It's ...
3
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1answer
146 views

What is the Cartan matrix for a dihedral group?

Dihedral groups are Coxeter groups of type $I_m$, $m \geq 3$. The Coxeter matrix of $I_m$ is \begin{align} \left( \begin{matrix} 1 & m \\ m & 1 \end{matrix} \right). \end{align} When $m=3,4,6$...
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2answers
146 views

Reference request: from a reduced expression of an element in a Coxeter group to another reduced expression

Are there some references which proves the following result? Let $W$ be a Coxeter group and $w \in W$. Then different reduced expressions of $w$ can be transformed from one into anther using only the ...
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1answer
105 views

Positive roots and elements in a Coxeter group.

In the paper, a set $L$ associated to an element $w$ in a Coxeter group $W$ is defined as follows. Let $w=s_{i_1} \cdots s_{i_m}$ be a reduced expression. Define $L=\{\beta_1, \ldots, \beta_m\}$, ...
12
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0answers
261 views

A hard Lefschetz theorem for nilCoxeter algebras

Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter algebra (over the reals, say), as defined at https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has a natural ...
2
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2answers
233 views

About reflections of reflection groups

For any finite crystallographic reflection group $W = \langle s_1, \ldots , s_n\rangle$, every hyperplane reflection is of the form $ws_iw^{-1}$ for some $i$ and some $w \in W$. A finite ...
3
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1answer
150 views

Injection from Artin monoids to Coxeter groups

Let $\sigma$ be a permutation. If two positive braids represent $\sigma$ and are of minimal length among the braids representing $\sigma$, then they are equal. From what I could gather, this result is ...
61
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0answers
2k views

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 combinatorial ...