12
votes
Finite groups with bounded centralizers
There is a paper by Daniel Palacin ("Finite groups contain large centralizers", Israel Journal of Mathematics, 244,(2), (2021), 621-624) which proves (without the classification of finite ...
- 44.4k
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}}
\...
- 14.1k
7
votes
Accepted
Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group
This is true for $p$ not a torsion prime of $G$; it is exactly Theorem 3.14 (p. 88) of Steinberg - Torsion in reductive groups (MSN). In fact, as you probably know, it's always good to look in ...
- 12.9k
6
votes
Coordinate ring of universal centralizer (BFM space)
For concreteness, let me provide an explicit description of the ring in Roman's answer above for $G=GL(n)$. There are several versions of universal centralizers in their paper, and I will describe two ...
6
votes
Accepted
Coordinate ring of universal centralizer (BFM space)
$\DeclareMathOperator\Spec{Spec}$I admit there is a mistake in our paper. It can be corrected though in a reasonably straightforward way, so the main results of the paper hold. We will post a ...
- 1,526
5
votes
Accepted
Centralizers in semisimple Lie group
The condition is known as "strongly regular" (and I think that the modern usage is "regular semisimple", hence also "strongly regular semisimple", not just "regular&...
- 12.9k
4
votes
Accepted
$|C(E):C(E)\cap C(Z(U))|=1$ or $p$
One possible argument is to compare this with $T/C(Z(U))$ as follows. We have $$C(E)/C(E)\cap C(Z(U))\cong C(E)C(Z(U))/C(Z(U)) \leqslant T/C(Z(U))$$ Now $T/U$ is cyclic, so $T/C(Z(U))$ is cyclic. It ...
- 16.2k
3
votes
Accepted
normalizer quotient is $\operatorname{GL}_2(p)$
Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&...
- 16.2k
3
votes
Accepted
Centralizers in Jacobson-Witt Lie algebras
For $W(n,1)={\rm Der}(\mathcal{O}_n)$, defined over an algebraically filed $k$ of characteristic $p>2$, the smallest dimension of centralizers equals $n$ (here $\mathcal{O}_n$ is the $k$-algebra $k[...
- 2,624
3
votes
Accepted
Centralizers of semisimple subgroups
In characteristic zero, the connected closed subgroups of $G$ are in 1-1 correspondence with the Lie subalgebras of ${\rm Lie}(G)={\mathfrak g}$, and the Killing form a suitably chosen symmetric ...
- 1,336
2
votes
Centralizer of a cyclic subgroup within the group algebra $\mathbb{C} S_N$ of the symmetric group
$\newcommand{\IC}{\mathbb{C}}$
Let $V_\lambda$ be the irreducible $S_N$-module (Specht module). The representations $\rho_\lambda: \IC S_N \to \operatorname{End}_\IC(V_\lambda)$ give us an isomorphism ...
- 9,650
2
votes
Centralizer of a cyclic subgroup within the group algebra $\mathbb{C} S_N$ of the symmetric group
Thanks to Mark Wildon and the OP for pointing out that my answer was incorrect- I was considering the centralizer of $\sigma$ in the wrong algebra.
However, it does seem to me that the structure ...
- 44.4k
1
vote
Accepted
An explicit matrix form
It looks like you are working with respect to the orthogonal form with matrix $\begin{pmatrix} & w_0 \\ w_0 \end{pmatrix}$, where $w_0 = \operatorname{antidiag}(1, \dotsc, 1)$. That's the one ...
- 12.9k
1
vote
Accepted
action of the extra-special group
Take $T_4$ as the diagonal torus, which is composed by 4-tuples of 2x2 rotation matrices $a,b,c,d$: $$\begin{bmatrix}a & & & \\ & b & &\\ & & c &\\ & & &...
- 9,501
1
vote
Accepted
A group-theoretic lemma in a paper by Ershov and He
Here is @AndyPutman's comment as an answer (so that it can be accepted), made CW to avoid reputation. If @AndyPutman prefers to post the answer, then I will delete this.
This only needs the fact ...
1
vote
Accepted
Intersection of identity components
$\DeclareMathOperator\Cent{C}\newcommand\oG{\overline G}\newcommand\oe{\overline e}$Put $\oG = \operatorname{PGL}_n(\mathbb C)$. I hope you will permit me to denote the involutions by $\oe_i$ instead ...
- 12.9k
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