14
votes
Accepted
Are (semi)simple Lie groups some sort of "homotopy quotient groups" of their maximal tori?
Here's the "answer" that I started writing, then put away for a while. The short answer is: although "T//W" is not the same as G, they "look sort of the same" from the point of view of certain ...
- 26.7k
12
votes
Accepted
One element commutation classes of reduced decompositions of the longest element of the Weyl group
The reduced words that are in their own commutation classes are:
$s = [123\cdots(n-2)(n-1)(n-2)\cdots321][23\cdots(n-3)(n-2)(n-3)\cdots32][3\cdots(n-4)(n-3)(n-4)\cdots3]\cdots$
the reversal of $s$ (...
- 136
11
votes
Accepted
Instructions for using Coxeter 3.0 software
If you type "help" immediately on entering the program, you'll get a fairly long and useful introductory message. At whatever level you are, you are supposed to be able to type "help," and then the ...
- 541
10
votes
Accepted
What is known about finite dimensional modules over the nilCoxeter algebra?
This algebra has just one isomorphism class of simple module - let's call it $S$. Its projective cover is the regular representation, and is also the injective hull. The socle of the regular ...
- 11.8k
9
votes
Accepted
$N_{G}(E)/C_{G}(E)$ is the Weyl group of $G$?
$\newcommand{\ZZ}{{\mathcal Z}_G}
\newcommand{\NN}{{\mathcal N}_G}
\newcommand{\zz}{{\mathfrak z}_G}
\newcommand{\Lie}{{\rm Lie\,}}
\renewcommand{\tt}{{\mathfrak t}}
\renewcommand{\gg}{{\mathfrak g}}
\...
- 13.6k
9
votes
Accepted
Invariants of cohomology of Springer sheaf
You want to look at the partial Grothendieck-Springer resolution, i.e. the variety of pairs $ (g \in G/ P_\mu, v \in g \mathfrak p_\mu g^{-1})$.
The partial Grothendieck-Springer resolution is smooth, ...
- 137k
9
votes
One element commutation classes of reduced decompositions of the longest element of the Weyl group
I believe that for $n \geq 4$ there will be exactly $4$ such reduced words. One such word, call it $R_n$ can be constructed by starting with $s_{n-1}s_{n-2} \cdots s_2s_1s_2 \cdots s_{n-2}s_{n-1}$ and ...
- 2,693
8
votes
Accepted
When the longest element of Weyl group is rational?
Let $B$ be a Borel subgroup containing $T$. As $F(B)$ and $B$ are both Borel subgroups containing $T$ there exists an element $n \in N_G(T)$ such that ${}^nF(B) = B$. Thus the Frobenius endomorphism $...
- 2,842
8
votes
Accepted
Each $w\in W$ can be expressed as product of distinct reflections?
The answer is "yes" and there is a geometric explanation.
Let $\mathcal{H}$ denote the set of hyperplanes corresponding to the reflections $s_\alpha$ with $\alpha\in\Phi^+$ (note that $s_\alpha=s_{-\...
- 2,846
7
votes
Accepted
Weyl group action on maximal tori
The first statement amounts to determining whether $T^w$ is connected. In the simplest case $G={\rm SL}_2$, $T$ may be identified with the multiplicative group ${\rm G}_m=\Bbb{C}^{\times}$, the Weyl ...
- 14.3k
7
votes
Cells in affine Weyl groups
You are asking a number of related questions here, most of which require more reading of Lusztig's papers. See the reference list in my conference paper here, for example. But note first that the ...
- 52.4k
7
votes
Accepted
Fixed space of maximal torus and Weyl group
This paper of Humphreys addresses your second question (the first is answered in the comments - the $W$-module structure is independent of the choice of torus): https://people.math.umass.edu/~jeh/pub/...
- 6,654
7
votes
Number of reduced decompositions of the longest element of the Weyl group
Around the time Fomin and I wrote this
paper, Tao Kai Lam applied the technique to type $D_n$. It
emerged that it was "natural" to weight a reduced
decomposition $\rho$ by $2^{d(\rho)}$, ...
- 49.5k
6
votes
Accepted
If a Weyl element preserves a root, then it has a representative which preserves the root space?
This is true if $G$ is split. First of all $w$ is represented by some $\tilde w\in N_G(T)\cap G(k)$ (see Borel-Tits, for example). Then $\text{Ad}\, \tilde w$ acts on $\mathfrak g_\alpha$ by some ...
- 15.3k
6
votes
Accepted
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
Expanded with lots of references at @JimHumphreys's request. Borel is Borel - Linear algebraic groups (2nd edition). Springer is Springer - Linear algebraic groups (2nd edition).
Let $G$ be ...
- 11.4k
6
votes
Accepted
Parabolic subgroup of Weyl group
No. In $A_2$ with $I=\{s_1,s_2\}$, take $J=\{s_1\}$ and $u^J=id$ and $v^J=s_1s_2$. Then $M=W_J$ because $s_1\le s_1s_2$.
- 1,816
6
votes
Accepted
What is the effect of tensoring with the sign representation on irreducible modules for a Type D Weyl group?
Recall the construction of such representations. Let $H\subset S_{2n}$ be the subgroup fixing the partition (so the two parts can be swapped)
$$\{1,\dots,n\}\sqcup\{1',\dots,n'\}.$$
It is isomorphic ...
- 1,897
5
votes
Accepted
Characterization of all $w$ in the Weyl group satisfying $w \geq w_l w_{l, \theta}$
Yes, this follows from the fact that $x \mapsto w_l x$ is an antiautomorphism of the Bruhat order on a finite Coxeter group. (See Björner and Brenti, Proposition 2.3.4, for example, but their $w_0$ ...
- 1,816
5
votes
Accepted
Parabolic Kazhdan-Lusztig polynomial coincide?
Yes, that's true. The standard recursive constructions will give you this fact easily, because the only group elements involved in $P_{x,w}^I$ are those which are $\leq w$ w.r.t. the Bruhat order. If $...
- 9,539
5
votes
Conjugacy of Regular Semisimple Subalgebras
It would help to recall Dynkin's definition of "regular semisimple subalgebra" of a complex semisimple Lie algebra (say $\mathfrak{g}$), which has not become standard in later literature. This just ...
- 52.4k
5
votes
Accepted
Unicity of the BGG complex
Theorem 33 in the preprint [1] gives the uniqueness of BGG resolutions (= direct sums of Verma modules resolving a simple module) in category $\mathcal{O}$, both in regular and singular blocks. (...
- 1,368
5
votes
Accepted
Number of reduced decompositions of the longest element of the Weyl group
This is easy to do in SageMath. E.g. the following code
G = WeylGroup("F4")
w = G.long_element_hardcoded()
print(w)
rw = w.reduced_words()
len(rw)
...
- 8,157
5
votes
Accepted
Particular reduced expression of the longest element of Weyl group
Whenever $\ell(wv)=\ell(w)+\ell(v)$, you can construct a reduced word for $wv$ by producing one for $w$ and one for $v$ and then concatenating them. So, if you know an algorithm for producing reduced ...
- 44k
5
votes
Accepted
Describing characters of a reductive group in terms of characters of a maximal torus
I thought at first that you meant "trace character of a finite-dimensional, irreducible representation of $G$" by "character of $G$". Then this is false; for example, if $G$ is $\...
- 11.4k
4
votes
Weyl group elements fixing a set of simple roots
I'm not sure whether there is an efficient way to answer your question (or a written reference), but it's possible to analyze the situation case-by-case.
It's probably best to start with an ...
- 52.4k
4
votes
Length of Weyl group element mapping highest root to a simple root
Ok I'm late on this, but I've just seen the question! Another way to look at it: the W-orbit of the highest root consists of the long roots, so the long roots are in bijection with the minimal coset ...
- 793
4
votes
Diagonal automorphisms for twisted Chevalley groups
First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include ...
- 52.4k
4
votes
Accepted
Weyl Group Element $w$ fixing a root, and its presentation as product of simple reflections $w=s_1\dots s_n$
This is definitely not true. For instance, already in $\Phi=B_2$, each root has a root orthogonal to it, so for every root there is some nontrivial element (in fact, a reflection) of the Weyl group ...
- 22.9k
4
votes
Each $w\in W$ can be expressed as product of distinct reflections?
After reading your comments, I come up with the following proof. I would like to know whether my proof is correct or not.
For weyl group $W$, each $w\in W$ can be expressed as $w=s_{\beta_l}\cdots s_{...
- 1,855
4
votes
Accepted
Bruhat order and positive roots made negative
There is a counterexample in $\mathfrak{sl}_3$. Denote by $\alpha, \beta$ the simple roots, and $s,t$ the corresponding simple reflections. Then $\Phi_s^- = \{\alpha\}$ and $\Phi_{st}^- = \{\beta, \...
- 1,368
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