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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Definitions of Hecke algebras
There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are ...
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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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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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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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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 $\mathfrak{h}$...
- 8,707
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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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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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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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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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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 system automorphisms as inner automorphisms of extended Chevalley group
For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...
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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 ...
- 8,707
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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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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 ...
- 8,707
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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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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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