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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What is known about finite dimensional modules over the nilCoxeter algebra?
Recall that the nilCoxeter algebra $\mathcal{N}_W$ for a Coxeter group $W$ is given by the $\mathbf{k}$-basis $x_w$ for each $w\in W$ and multiplication $x_ux_v=x_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$ ...
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Chevalley-Solomon formula and Weyl character formula
Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
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Root systems of Weyl groupoids
I am working with the notion of Weyl Groupoids as introduced in "A generalization of Coxeter groups, root systems, and Matsumoto’s theorem" by Heckenberger and Yamane.
The authors generalize ...
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When is an affine left cell finite?
Consider an affine Weyl group $\hat W$ of a simple Lie type. Let $w \in \hat W$ and let $C^L(w)$ denote the left cell in $\hat W$ containing $w$.
Is there a good criterion to test whether $C^L(w)$ has ...
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A combinatoric identity for characters of reductive groups
Let $G$ be a reductive group over an algebraic closed field (of char 0 if necessary). Let $T\subset G$ be a maximal torus and $S=\mathrm{Sym}^*(X(T))$ be the symmetric algebra of characters of $T$. ...
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What is the effect of tensoring with the sign representation on irreducible modules for a Type D Weyl group?
Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...
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When are these irreducible complex representations for the Type D Weyl group self-dual?
Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...
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Elements of length 0 in extended affine Weyl group for GL(n)
As part of my research, I would like to understand the possible pairs of $(v,\sigma)\in \mathbb Z^n\times S_n$ satisfying the following condition: For $1\le i < j \le n$, we have $\sigma(i) < \...
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Stabilizer of a Levi subgroup in the Weyl group and its quotient
(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.)
For simplicity, let $G$ be a connected reductive ...
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Weyl groups are Coxeter groups proof
I'm reading part of a proof that says that Weyl groups of apartments of buildings are Coxeter groups.
Let $\Delta$ be a building and let $\Sigma$ be a fixed apartment of $\Delta$. Let $C$ be a fixed ...
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Cohomology of Deligne-Lusztig variety associated to Coxeter element
Determining the individual ($l$-adic) cohomology groups of Deligne-Lusztig varieties has only been done for the general linear group and for some other very specific cases (as far as I know).
However, ...
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Minimal subrepresentation of the Weyl group
Background
I want to generalize Theorem 3.2.1 in
Dat, J., Orlik, S., & Rapoport, M. (2010). Period Domains over Finite and p-adic Fields (Cambridge Tracts in Mathematics). Cambridge: Cambridge ...
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Does -I belong to Weyl group?
Let $\Phi$ be an irreducible root system, with positive roots $\Phi^+$ relative to the base $\Delta$.
If $W$ is the Weyl group, how can I determine if $-I$ belongs to $W$? Equivalently how can I see ...
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Representations of $\mathrm{sl}(3,\mathbb{C})$ and Catalan-like paths
Background on representations of $\mathrm{sl}(3,\mathbb{C})$
In Chapter 6 of Brian C. Hall's book "Lie Groups, Lie Algebras, and Representations", he constructs the irreducible ...
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Describing characters of a reductive group in terms of characters of a maximal torus
Say I have a reductive complex algebraic group $G$ with maximal torus $T$ and associated Weyl group $W$. I would like to be able to say that the characters of $G$ are in bijection with the $W$-...
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Weyl group action on the Lie algebra [duplicate]
Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...
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Relative position of flags for the general linear group
This question is related my question in An example of a Deligne–Lusztig variety for a general linear group, and the obtained answer.
Situation
I am working with the general linear group. Specifically, ...
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Is this construction related to the geometric Langlands program perhaps?
Given a complex Lie algebra $\mathfrak{g}$, a choice of Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and a dominant integral weight $\lambda$ of $\mathfrak{g}$, there is a natural construction ...
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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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One element commutation classes of reduced decompositions of the longest element of the Weyl group
For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the ...
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Sum of weights of an irreducible representation of $U(N)$
Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries.
Firstly, I would like to know ...
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Do Weyl groups generate the exceptional Lie groups as sequences of reflexions in the Weyl chambers?
Platonic groups of symmetry are Weyl groups for the exceptional Lie algebra E6->E8, as root systems. These can be viewed as mirrors in a kaleidoscope (Goodman).
I would like to know if one can ...
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Real roots along root strings
Let $A$ be a Cartan matrix, i.e. a $n\times n$ matrix with integer entries such that $A_{ii}=2$ and $A_{ij}\leq0$ for $i\neq j$. Then the corresponding Kac-Moody Lie algebra has a Cartan subalgebra $\...
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Question on the representative of the longest Weyl element of $\mathrm{SO}(2n+1)$
Let $w_{m}$ be the $m \times m$ matrix with ones on the non-principal diagonal and zeros elsewhere.
Let $V$ be the $2n+2$-dimensional quadratic space with the symmetric bilinear form $\left<,\right&...
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A construction of Weyl-equivariant maps from the space of regular Cartan triples to the space of tuples of complex polynomials (up to scalar factors)
Let $G$ be a compact semisimple Lie group and let $T$ be a maximal torus in $G$. On the Lie algebra level, we have a real Lie algebra $\mathfrak{g}$ and a (particular) real slice, say $\mathfrak{t}$, ...
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elementary abelian subgroups with centralizers not connected
Let $G =$ PGL$_{8}(\textbf{C})$. Let $a, b, c, d$ be four representatives of conjugacy classes of involutions in $G$ where $$a = \begin{pmatrix}
-1 & 0\\
0 & I_{7}
\end{pmatrix}, b = \begin{...
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Particular reduced expression of the longest element of Weyl group
Let $I$ be the Dynkin diagram vertex set and $K$ be a proper nonempty subset of it. Let $w_0^K$ be the longest word of the Dynkin subdiagram $K$, which might be a disjoint union of connected Dynkin ...
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Invariants of cohomology of Springer sheaf
Let $G=Gl_n(\mathbb{C})$ and $\mathcal{N}$ be the nilpotent cone associated to it i.e nilpotent matrices inside $\mathfrak{g}=\mathfrak{gl}_n(\mathbb{C})$.
We have the variety $\tilde{\mathcal{N}}$ ...
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Parabolic subgroup of Weyl group
Let $W$ be the Weyl group of a semisimple algebraic group $G$. $I$ be the simple roots. $J\subset I$ generate a parabolic subgroup of $W$ denote by $W_J$. $w^J$
is the shortest representative of $w$ ...
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$N_{G}(E)/C_{G}(E)$ is the Weyl group of $G$?
In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ denote the subgroup of elements of order dividing 2 in the diagonal maximal torus; it is generated by the images of the three ...
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Polynomial invariants of the exceptional Weyl groups
Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
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Action of Weyl group on regions of Shi arrangement
This is an elaboration of a question which was aked on MO several years ago, which was unanswered but deleted by the question-asker. I hope it is okay to elaborate on a deleted question like this; for ...
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Weyl group and Galois action on cubic surfaces
Let $X$ be a smooth cubic surface over a field $k$. Denote by $\bar{k}$ the separable closure of $k$ and $\bar{X}:=X\times_{k}\bar{k}$. Then it is well know that there exists a homomorphism
$$
\phi:\...
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Geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group
What is the geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group? Could Springer theory of Weyl group representations be used to obtain such a geometric meaning?
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The Weyl group of Kac-Moody algebra and Coxeter group
Let $\mathfrak{g}$ be a Kac-Moody algebra, $W$ be its Weyl group, generate by fundmental reflections {$r_1,...,r_n$}. For $i,j$, we have relation $(r_ir_j)^{m_{ij}}=1$, $1\le i,j\le n$. Let $W^{'}$ be ...
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Positive roots and the longest element of the Weyl group
Take $\frak{g}$ a complex semisimple Lie algebra and its Weyl group $W$. Is it true that the number of positive roots of $\frak{g}$ is equal to the length of the longest element of $W$?
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How to determine sublattices S of a root lattice R
Let $R$ be a root lattice of a irreducible root system $\Phi$.
Suppose $W$ is a Weyl group of $\Phi$ and $S$ is a sublattice of $R$ which is $W$-stable and satisfies $|R/S|<\infty$.
For example, ...
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Reference request: Weyl group action on the power set of positive roots
There is a symmetric group action on the power set of positive roots in type A. The action is defined as follows.
Denote by $\alpha_1, \ldots, \alpha_n$ be the set of simple roots in a root system. In ...
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Action of Coxeter element on mod $2$ root lattice is semisimple
Let $\Lambda$ be a simply laced root lattice and $w$ a Coxeter element of the Weyl group of $\Lambda$.
Question: Is it true that the action of $w$ on the $\mathbb{F}_2$-vector space $\Lambda/2\Lambda$ ...
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Condition for a sum of images of fundamental dominant weights to lie on a wall
Let $\Delta$ be a system of simple roots in a root system with Weyl group $W$. For $\alpha\in\Delta$, let $\varpi_\alpha$ be the corresponding fundamental dominant weight. Let $w\neq r$ be elements of ...
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Conjugation of root subgroups by the Weyl group
Fix a field $k$ of characteristic zero, and let $G$ be a connected reductive algebraic $k$-group of isotropic rank $\ge 1$. Fix a maximal $k$-split torus $S$, and let $\Phi_k$ be the relative root ...
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Two-sided cells, special nilpotent orbits and special representations
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. This question concerns three classical objects of representation theory: the two-sided Kazhdan-Lusztig cells of the Weyl group $W$ of $\mathfrak{...
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Are there mathematical/physical applications of these Weyl equivariant maps?
Let $G$ be a compact Lie group and $T$ a choice of maximal torus. Denote the corresponding Lie algebras by $\mathfrak{g}$ and $\mathfrak{t}$. Elements of $\mathfrak{t} \otimes \mathbb{R}^3$ are called ...
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Name for an involution associated to a Coxeter element
Let $(W,S)$ be a finite Coxeter system, and $c \in W$ a Coxeter element.
There is an involution $g\in W$ for which the involutive map $w \mapsto gw^{-1}g$ fixes $c$. Is there a standard name for this ...
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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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Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?
There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
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Description of Soergel modules
So this is asking a basic and/or stupid question (my apology and appreciation) about Soergel modules that comes out of exercises by me who knows little about the subject.
Let $W$ be a finite Weyl ...
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On $\Psi$-generating paths in the Bruhat order of a Weyl group
Let $W$ be a Weyl group with roots $R$ and positive roots $R^+$. Let $v\in W$ of length $r$. We call $\mathbb{m}=(\alpha_1,\ldots,\alpha_r)\in(R^+)^r$ a Bruhat path from $1$ to $v$ if $1\lessdot s_{\...
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Number of reduced decompositions of the longest element of the Weyl group
Let $R$ be a reduced root system, $W$ the associated Weyl group, and $w_0 \in W$ the longest element of $W$. In general $w_0$ admits more than one reduced decomposition into a product of reflections, ...
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Uniqueness in Mare combinatorics and bounds on Gromov-Witten invariants
Let $R$ be the root system of a Weyl group $W$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. the set of positive roots $\alpha$ such that $\ell(s_\alpha)=2\operatorname{ht}\alpha-1$. Based ...
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