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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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 ...
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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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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 ...
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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.
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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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0
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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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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 ...
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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 ...
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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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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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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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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 ...
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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 ...
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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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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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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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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\...
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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) ...
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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 ...
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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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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?
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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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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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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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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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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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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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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$?
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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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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,$$
where we embed the ...
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Reference request: Coxeter length and irreducible characters
Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have
$$\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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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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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 ...
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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 ...
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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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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 ...
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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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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 ...
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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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$E_6$, $E_8$, 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 ...
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1
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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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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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1
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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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2
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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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1
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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 ...
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10
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1
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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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7
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1
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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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1
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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 lies in the centralizer of the longest word $w_0$ in $S_{2n}$.
Thank you very much.
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