All Questions
13 questions
3
votes
0
answers
195
views
A property of an irreducible root system
Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
- 673
3
votes
0
answers
202
views
The group of fixed points of an involution of a Weyl group
Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$.
Let $W=W(R)$ denote its Weyl group.
Let $S\subset R$ be a basis of $R$ (a system of simple roots).
Let $D=D(R,S)$ denote the ...
- 14.2k
6
votes
1
answer
542
views
Is the connected centralizer of a semisimple element in a connected reductive group also a centralizer?
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field and let $g\in G$ be semisimple. Write $C=\mathrm{C}_G(g)$ and $C^\circ=\mathrm{C}_G(g)^\circ$ for the ...
- 993
2
votes
1
answer
304
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 ...
- 332
1
vote
0
answers
112
views
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. ($x_{i,...
- 332
8
votes
1
answer
2k
views
Does the classification of reductive groups follow from that of semisimple groups?
I had a question for anyone familiar with the proofs of the classification of reductive groups. I skipped most of the details of classification when I originally learned linear algebraic groups, and ...
- 6,180
4
votes
1
answer
255
views
How to tell when two abstract root data are isomorphic
This is a related question to one I just asked ($\textrm{GSp}_{4}^{\wedge} \cong \textrm{GSp}_4$).
Let $\Psi=(X,R,X^{\wedge},R^{\wedge}), \Psi_1 = (X_1,R_1,X_1^{\wedge},R_1^{\wedge})$ be two root ...
- 6,180
5
votes
2
answers
625
views
$\textrm{GSp}_{4}^{\wedge} \cong \textrm{GSp}_4$
Everything is over an algebraically closed field of characteristic $\neq 2$. I had constructed a root datum for $\textrm{GSp}_4$ in a less than ideal way, and I had hoped to show that it was self ...
- 6,180
8
votes
2
answers
1k
views
Definition of $\textrm{GSpin}_{2n}$ and its root datum
I'm trying to get my hands on the general spin group $G = \textrm{GSpin}_{2n}$. It have seen it mentioned as a connected, reductive group whose derived group is $\textrm{Spin}_{2n}$, which is the ...
- 6,180
6
votes
1
answer
186
views
Is $\overline{\Delta}$ a linearly independent set?
Let $G$ be a connected reductive group defined over a field $F$. Let $T$ be a maximal torus of $G$ which is defined over $F$, $A_0$ the maximal $F$-split subtorus of $T$, and $X_0$ the cotorsion free ...
- 6,180
5
votes
1
answer
453
views
A subgroup of the Weyl group
Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2.
Let $Q=Q(D)$ denote the root lattice of $D$.
Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is ...
- 14.2k
4
votes
1
answer
282
views
Name for a class of parabolic subgroups
This is a reference request for a (the) name of the following class of parabolic subgroups of a complex simple Lie group $G$:
Recall that parabolic subgroups of $G$, containing fixed Borel subgroup, ...
- 31.2k
3
votes
0
answers
289
views
Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups
If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
- 31