All Questions
8 questions
5
votes
0
answers
231
views
Avoiding Cartan subalgebra in a Lie algebra
Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation.
What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ ...
- 309
6
votes
1
answer
317
views
Action of complex torus on a vector space
Consider a torus $T$ over $\mathbb{C}$. Let $\rho: T\rightarrow \operatorname{GL}_{n}(\mathbb C)$ be a finite dimensional complex representation.
Is there an elementary way (undergrad level) to see ...
- 3,472
2
votes
1
answer
168
views
Irreducible $G$-representations with unital algebra structure
Let us work over $\mathbb C$. Suppose that $G$ is a semisimple algebraic group and let $H \subset G$ be a maximal torus. Consider a dominant weight $\omega$, then one can associate a unique ...
- 381
3
votes
1
answer
237
views
invariant subspaces of general linear groups for finite fields
Let $K$ be a finite field, let $n\ge 1$ be an integer and let $G=\mathrm{GL}(n,K)$ be acting linearly on a finite dimensional $K$-vector space $V$. Although $G$ is a reductive group, it is not ...
- 7,722
4
votes
1
answer
298
views
Characterizations of groups whose general linear representations are all trivial
Let $G$ be a group. Suppose for any general linear representation $\rho:G\to\mathrm{GL}(n)$,
$\rho$ must be trivial.
Question: Are there any characterizations or equivalent conditions for $G$?
Thanks ...
- 1,990
1
vote
0
answers
64
views
Conjugacy classes and normal form of $O_n$ and $U_n$
I'm interested in characterizing conjugacy classes inside $O_n$ and $U_n$ over local fields of positive characteristic ($\neq 2$). I need this for my research on representation theory of these groups.
...
- 11
6
votes
1
answer
273
views
Simultaneous triangularisation of an exterior power of a set of matrices
I'm working on some research problems relating to random matrix products, and this is taking me into areas of mathematics I've not previously studied: Lie groups, representation theory, and real ...
- 6,206
3
votes
2
answers
707
views
$k$ structures on $K$ vector spaces
The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.
Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is ...
- 1,553