All Questions
Tagged with lie-groups weyl-group
30 questions
1
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0
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46
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The difference between two description of affine Weyl groups
I have a question about the difference between two description of affine Weyl groups.
Let me write two descriptions of affine Weyl groups:
Let $\mathfrak{g}=\mathfrak{g}(A)$ be affine Lie algebras ...
- 137
3
votes
1
answer
129
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Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group
When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis by Beck, Chari, and Pressley, and I have some questions about some notations.
In the paper, we assume that $\...
- 137
12
votes
3
answers
795
views
The orders of the exceptional Weyl groups
Who first calculated the orders of the Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$? How were these orders calculated?
- 417
7
votes
1
answer
259
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A name for the Weyl group of $\frak{so_{2n}}$
For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.
A) Does the $D$-series Weyl group $S_n \...
- 417
1
vote
0
answers
113
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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 $T$. ...
- 11
1
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0
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105
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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 ...
0
votes
1
answer
194
views
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 ...
- 111
2
votes
2
answers
353
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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 ...
- 201
2
votes
0
answers
213
views
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 $W$?
- 393
4
votes
0
answers
184
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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 ...
- 5,215
9
votes
2
answers
657
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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, ...
- 219
6
votes
1
answer
216
views
Fixed space of maximal torus and Weyl group
Let $G$ be a compact connected Lie group and $T\subset G$ a maximal torus. Let $V$ be a representation of $G$ and $U=\{v\in V: tv=v\textrm{ for all }t\in T\}$. For any $g\in N(T)$ we have for all $t\...
- 3,031
3
votes
0
answers
133
views
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 ...
- 31
2
votes
1
answer
304
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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 ...
- 332
10
votes
0
answers
392
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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.
...
- 24.2k
1
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0
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112
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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,...
- 332
1
vote
0
answers
370
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(\...
- 332
2
votes
1
answer
217
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/...
- 235
26
votes
1
answer
940
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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 ...
- 18.4k
1
vote
1
answer
528
views
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 $...
- 125
3
votes
1
answer
396
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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 $k$. ...
- 6,180
2
votes
1
answer
169
views
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 $\...
- 960
7
votes
2
answers
508
views
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 ...
- 2,995
5
votes
1
answer
2k
views
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 ...
- 951
4
votes
1
answer
516
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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 ...
- 4,729
4
votes
0
answers
119
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Topological obstruction to icosahedral symmetry?
Let $G$ be a compact simple lie group of rank $n$. Then the Poincaré series of $G$ is given by $$P(G,q)=\prod_{i=1}^n (1+q^{2d_i-1}),$$
where the integers $d_1\leq d_2\leq \cdots \leq d_n$ are the ...
- 3,761
15
votes
2
answers
764
views
Is there a category of representations of a simple Lie algebra on which its Weyl group naturally acts?
For any simple Lie algebra $\mathfrak{g}$, is there a category $C$ of (possibly infinite-dimensional) representations of $\mathfrak{g}$ such the Weyl group $W$ of the corresponding root system acts in ...
- 22.3k
2
votes
1
answer
223
views
The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular
This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We ...
- 9,019
16
votes
5
answers
2k
views
About the intrinsic definition of the Weyl group of complex semisimple Lie algebras
It may be a easy question for experts.
The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$...
- 9,019
20
votes
6
answers
4k
views
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}...
- 2,160