# Questions tagged [weyl-group]

The weyl-group tag has no usage guidance.

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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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### odd positive roots of basic classical Lie superalgebras

In the Lie algebra case, positive roots are "almost ($S_{\alpha}$ permutes $\Delta^+ -\{\alpha\}$)" invariant under simple reflections. A similar statement I want to understand for Lie superalgebras.
...

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### Comparison of length functions on Weyl groups

Let $G$ be a connected reductive group over an algebraically closed field $k$ (with nice enough characteristic), and let $\sigma:G\to G$ be a finite order automorphism of $G$. The connected component $...

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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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### Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra

Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-...

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### About weight of $M\in\mathcal{O}_{\chi_\lambda}$ in the Category $\mathcal{O}$

Given $M\in\mathcal{O}_{\chi_\lambda}$, how does the weights of $M$ relate to $\lambda$? Is it something about $W\cdot\lambda$?

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### About block of category $\mathcal{O}$

In the book "Representation of Semisimple Lie Algebra in the BGG Category $\mathcal{O}$".
Exercise 1.13. Suppose $\lambda\not\in\Lambda$, so the linkage class $W\cdot\lambda$ is the disjoint union of ...

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### Relationship bewteen Kazhdan-Lusztig Vogan polynomial and Kazhdan-Lusztig polynomial

Let $W^I=\{w\in W: w^{-1}\Phi_I^+\subseteq \Phi^+\}$, $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$
Let $M(\lambda)$ be the Verma module with highest weight $\...

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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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### Each $w\in W$ can be expressed as product of distinct reflections?

For a Weyl group $W$, I would like to know whether each $w\in W$ can be expressed as $w=s_{\alpha_1}s_{\alpha_2}\cdots s_{\alpha_k}$ for some distinct positive roots $\{\alpha_1, \alpha_2, \cdots, \...

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### Combinatorial and computational problem related to Weyl groups and the coroot lattice

Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\...

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### What is the longest word in type $C_2$ Weyl group written in terms of a matrix?

The longest word in type $A_3$ Weyl group written as a matrix is
\begin{align}
w_0=s_1s_2s_1=\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & ...

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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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### Why the hyperoctahedral group is a ``reductive'' group?

Sorry for the misleading title, I actually mean the following:
The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...

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### Cohomology of root data

Let $(X,\Phi,X^\vee,\Phi^\vee)$ be a semisimple root datum (in the sense of SGAIII), and $W_0$ its (finite) Weyl group. What is known about the cohomology groups $H^n(W_0, X^\vee)$ ?

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### Are (semi)simple Lie groups some sort of “homotopy quotient groups” of their maximal tori?

Warning: non-specialist writing, some rubbish possible.
The formula $h^*(BG)\cong h^*(BT)^W$ valid for complex oriented cohomology of the classifying space of a compact Lie group $G$ with maximal ...

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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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### Conjugacy of Regular Semisimple Subalgebras

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. In his paper Semisimple Subalgebras of Semisimple Lie Algebras, Dynkin states that by a result of Weyl,
1) Two regular semisimple subalgebras ...

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### Weyl group action on maximal tori

Let $G$ be a semisimple algebraic group over the complex numbers and we fix a maximal torus $T$. Let $w\in W$ be an element in the Weyl group, and let $T^{w}$ be the elements in $T$ that are fixed by $...

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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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### The double cover of $[W(E_7),W(E_7)] \cong Sp_6(\mathbb F_2)$ as a Galois group over $\mathbb Q$

I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$:
Is the double cover of $Sp_6(...

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### Description of the center of a reductive group using absolute and relative roots

Let $G$ be a connected, reductive group over a field $k$. Let $T \subseteq B$ be a maximal torus and Borel subgroup of $G$ with corresponding base $\Delta \subseteq X(T)$. Then $T$ contains $Z(G)$, ...

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### Choosing canonical representatives of Weyl group elements, some questions

Let $G$ be a connected, reductive group which is quasisplit over a field $k$ of characteristic zero. Let $B$ be a Borel subgroup defined over $k$, containing a maximal torus $T$ defined over $k$. ...

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### 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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### A corollary to Weyl's formula on roots of Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra, and $W$ its Weyl group. We make a choice of simple roots, and note $\Phi_+$ the corresponding set of positive roots, while $\Phi$ is the set of all ...

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### How do I obtain Vandermonde identity from Weyl's denominator formula?

Let $\Phi$ be a root system with Weyl group $\operatorname{Weyl}(\Phi)$, let $\Phi^+$ be a set of positive roots for $\Phi$ and $\rho$ be the half sum of the elements of $\Phi^+$. Then the Weyl's ...

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### When the longest element of Weyl group is rational?

Let $\mathbb{G}$ be a connected reductive group over $\mathbb{F}_q$, and let $G$ be the base change to an algebraic closure of the base field. Denote by $F$ the associated geometric Frobenius.
Let $...

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### Weyl groups of Levi factors and their cosets

Let $G$ be a connected semisimple algebraic group defined over an algebraically closed field $F$ of characteristic $0$. Let $B$ be a minimal parabolic subgroup of $G$ (i.e. a Borel subgroup), let $P$ ...

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### Regarding Weyl's integration formula - from “equality up to scalar” to “exact equality”

Teaching a course on representations of compact groups, I want to prove Weyl's integration formula. It seems to me that in a few sources, they prove the formula up to scalar, and don't even ...

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### The mod p reduction of Weyl groups and their polynomial invariants

Let $C$ be an $n\times n$ Cartan matrix, then the Weyl group $W\subset GL(n,Z)$.
Let $W_p\subset GL(n,Z_p)$ be the mod p reduction of $W$. I want to know whether there are known work on the structure ...

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### Weyl group elements fixing a set of simple roots

Suppose I have a root system $\Phi$ (of a semisimple Lie algebra) with a set of simple roots $\Delta$. I am interested in describing Weyl group elements $w$ preserving a given subset $\Delta'$ in the ...

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### Kazhdan Lusztig map and Richardson orbits

Let $g$ be a simple Lie algebra with Weyl group $W$. Kazhdan and Lusztig defined a map
$\Phi$: nilpotent orbits in $g$ $\rightarrow$ conjugacy classes in $W$.
Let $\eta_p$ be a Richardson orbit ...

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### Element of the Weyl group acting on co-roots

Let $W$ be the Weyl group of a semisimple algebraic group defined over say $\mathbb C$. We have the root space $\mathfrak a^*$ spanned by the roots $\{\beta \in \Delta\}$ and also by the weights $\{ \...

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### On pairs of roots which are orthogonal but not strongly orthogonal

Let $R$ be a reduced, irreducible, crystallographic root system with positive roots $R^+$ and simple roots $\Delta$. Let $W$ be the Weyl group of $R$. Let $(-,-)$ be a $W$-invariant scalar product on $...

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### If a Weyl element preserves a root, then it has a representative which preserves the root space?

Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\...

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### Cells in affine Weyl groups

This may sound like a very general question, which pretty much reflects my ignorance on the subject.
In the case of Weyl groups $W$, there is a notion of left/right/double cells, which is roughly ...

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### Computing Springer action on the homology of affine Springer fibers

Lusztig defined (in Sec. 5, also Sage) a Springer action of the affine Weyl group on the homology of affine Springer fibers (Iwahori one, i.e. in an affine flag variety). In the regular semisimple ...

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### Is there a geometric interpretation of the Hecke algebra of a Weyl group?

The Weyl group $W$ can be realized as a fundamental group of a topological space. So that its group algebra $\mathbb{Z}(W)$ has a geometric meaning. On the other hand, the Hecke algebra is a ...

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### Length of Weyl group element mapping highest root to a simple root

Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the ...

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### Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...

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### Is this characterization of (-1)-eigenspaces of the Weyl group of $E_6$ known?

I recently needed to know which circles $S$ in a maximal torus $T^6$ of the compact exceptional group $E_6$ yield one-dimensional subspaces $\mathfrak s$ of the Lie algebra $\mathfrak t^6$ that are ...

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### Root in positive Weyl chamber

Let $\mathfrak{g}$ be a complex simple Lie algebra. We fix a Cartan subalgebra $\mathfrak{t}\subset \mathfrak{g} $. Let $R\subset \mathfrak{t}^*$ the set of roots. We fix $\Pi\subset R$ the set of ...

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### Counting group invariants using Macdonald conjecture?

It is known from Dyson, Macdonald $et$ $al$ that the constant term asscoiated to the expansion of the following expression
\begin{equation}
\prod_{\alpha \in \Delta} (1-e^{\alpha})^k
\end{equation}
...

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### Counting points on Hessenberg varieties over a finite field

Let $G$ be an connected reductive group over finite field $k$. I will assume that $\text{char}(k)$ is very good for $G$ (or even larger, if preferred). Let $B\subset G$ be a Borel subgroup defined ...

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### Highest (short) roots and commutation relations in (twisted) DAHA

I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the ...

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### Weyl groups of $E_6$ and $E_7$

The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...

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### Double quotients of Coxeter groups have the chain property?

Let $(W,S)$ be a Coxeter group with Bruhat order $\leq$ and length function $\ell(w)$.
Definition: a subset $X \subset W$ has the chain property if whenever $x,y \in X$ with $x < y$, there ...

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### Calculation with weights of $E_6$

Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...

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### Weyl group of a K3 surface

I am wondering wether the action of the Weyl group $W_X$ of a K3 surface $X$ is transitive on the sets of curves of fixed genus.
Suppose $W_X$ is non-trivial. Given two curves $C,C'$ of genus $g\...

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### Finding particular reduced words for Weyl group elements

I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper Partial flag varieties and preprojective algebras by Geiss, Leclerc and Schröer. One ...