# Questions tagged [weyl-group]

The Weyl group of a root system is a subgroup generated by reflections through the hyperplanes orthogonal to the roots.

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### 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 ...

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### Weyl group stabilizer of semisimple element in adjoint group

Let $G$ be semisimple group over $\mathbb{C}$ of adjoint type. Let $T$ be a maximal torus, $s\in T$ semisimple element. Let $W$ be a Weyl group and $W(s)$ be a stabilizer of $s$ in $W$. I am ...

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### Equivalence of categories between the loop algebra of $sl_{n+1}$ and the affine Weyl group of $GL_\ell(C)$

In this paper here, Theorem 4.9 page 18, Charri and Pressley are claiming that there exists an equivalence of Categories between certain categories of the Lie algebra $\tilde{ \mathfrak g}=\mathfrak{...

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227 views

### Unicity of the BGG complex

A friend and I are writing a paper that uses the BGG resolution of $L(\lambda)$ (where $\mathfrak g$ is a semisimple complex Lie algebra, $\lambda \in P^+$ is a dominant integral weight, and $L(\...

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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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126 views

### When does the Kazhdan-Lusztig polynomial $P_{x,w}(q)$ not vanish at $q=1$?

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra. For any $\lambda\in \mathfrak{h}^{*}$ let $M(\lambda)$ and $L(\lambda)$ be the Verma module and the simple ...

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86 views

### Is one of the hyperplane partitions of a irreducible root system always generate the whole Weyl group?

Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots.
We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\...

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120 views

### Can we have a nontrivial division of a irreducible root system as the union of two closed sub-root systems?

The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the ...

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**1**answer

153 views

### The simple reflections of the Weyl group in $\operatorname{SO}_{2n}(\mathbb C)$

Let $W$ be the Weyl group corresponding to the maximal torus $diag(t_1, . . . , t_{n}, t^{−1}_n, . . . , t^{−1}_1)$ in a Borel group of $\operatorname{SO}_{2n}(\mathbb C)$.
What are the matrices ...

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186 views

### Can we have a nontrivial division of a irreducible root system as $\Phi=\Phi_{[\lambda]}\cup \Phi_{[\mu]}$?

Let $(\mathfrak{g},\mathfrak{h},\Phi)$ be a root system of a complex simple Lie algebra, where $\Phi$ is the set of all roots. For each $\alpha\in \Phi$, let $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$ be ...

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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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94 views

### Symmetry and factors of formal character of a complex simple Lie algebra

Let $\mathfrak{g}$ be a complex simple Lie algebra. We fix a root system of $\mathfrak{g}$. Let $\bf{\Delta}$ be the set of roots and we fix $\Sigma = \{\alpha_1, \dots, \alpha_N\}\subset \bf{\Delta} ...

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179 views

### Diagonal automorphisms for twisted Chevalley groups

Let $G$ be a Chevalley group over a field $k$ of characteristic $0$. We know that a diagonal automorphism $\phi_h$ of $G$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ ...

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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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### Bruhat order and positive roots made negative

Let $(\Phi, V)$ be a reduced root system with base $\Delta$ and Weyl group $W$. Let $\ell$ be the length function of $W$ with respect to the set of simple reflections $S = \{s_{\alpha} : \alpha \in \...

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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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### About the structure of set that parameterizes the set of simple modules in a block of $\mathcal{O}^\mathfrak{p}$

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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**1**answer

172 views

### Conjugacy classes of rational tori in Symplectic group

Rational conjugacy classes of Frobenius stable tori (in a finite group of Lie type) are in bijection with Frobenius-conjugacy classes of the corresponding Weyl group.When the group is the Symplectic ...

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141 views

### Bruhat cell of a Coxeter element

If $G$ is a complex Chevalley group and $H\leq G(\mathbb Z)$ dense in $G(\mathbb C)$, can I find $g\in H$ conjugated in $G(\mathbb Z)$ to an element in the Bruhat cell $BwB$ where $w$ represent a ...

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219 views

### Fake degrees: why coinvariant algebra and classical groups over finite fields?

Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated.
...

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### Nontrivial relations of the irreducible root systems

For the root system of the type $A_n$, the roots are $\alpha _{i,j}$, $1\le i\neq j\le n$, we have the nontrivial relations $(x_{i,j} (t), x_{j,k}(u)) = x_{i,k}(tu)$ if $i, j, k$ are distinct. ($x_{i,...

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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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136 views

### Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$

Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...

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121 views

### Weyl Group Action on Littelmann Paths

In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via
$$
\tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi)...

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### Denominator identity for Lie superalgebras

Let $\mathfrak g$ be a basic classic simple Lie superalgebra.
Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...

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81 views

### Distance between Verma modules in certain “strongly” standard filtrations

On p. 128 of the book: Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$.
I quote: "......Delorme arrives at
vanishing criteria for Ext$^n(\mathcal{O})$ which are more ...

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149 views

### On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not

In the paper: Pattern Avoidance and Rational Smoothness of
Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), https://www.sciencedirect.com/science/article/pii/...

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### Weyl Group Element $w$ fixing a root, and its presentation as product of simple reflections $w=s_1\dots s_n$

Let $\Phi$ be a root system and $\gamma \in \Phi$ a root. Let $W$ be the Weyl group and $\Delta$ a set of simple roots. Let $w \in W$ such that $w(\gamma)=\gamma$. Is it true that if $w=s_1\dots s_n$ ...

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### Weyl theorem - possible corollary - alternative characterization of projective representation of $Z_N\times Z_N$

For an integer $N$, let $\omega=e^{2i\pi/N}$ and $A$, $B$ be the clock and shift operators:
$A=\left(\begin{matrix}
1 & 0 & \cdots & 0 \\
0 & \omega & \cdots & 0 \\
\vdots &...

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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 $...

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### Two seemingly different definitions of a left cell

This is a question about two seemingly different notions of a left cell in a finite Weyl group and why they are the same. My question arose from reading a paper of W. McGovern titled "Left cells and ...

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### Can the Weyl orbits of fundamental weights tell us the Cartan matrix?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\Delta$ its root system contained in $\mathfrak{t}^{\vee}$ for a Cartan sub-algebra $\mathfrak{t}$ of $\mathfrak{g}$. Let $W$ be its Weyl group....

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### Associated subgroup of Weyl group

Let $\Phi$ be a root system.
For a weight $\lambda\in\mathfrak{h}^*$,
start by defining
$
\Phi_{[\lambda]}:=\{\alpha\in \Phi \ | \ \langle \lambda,\alpha^{\lor}\rangle\in\mathbb{Z} \}
$
and
$
W_{[\...

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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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158 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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220 views

### 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 ...