New answers tagged invariant-theory
4
votes
Accepted
Eigenfunctions of the Laplace–Beltrami operator on the coadjoint orbit of $\mathfrak{su}(n)$
For the generic coadjoint orbit as you require, i.e. the full flag manifold $\operatorname{SU}(n)/\text{(max torus)}$, the paper of Yamaguchi (1979) cited at this question has not only the eigenvalues,...
- 31.5k
14
votes
Example of non homogenous manifold with a finitely generated algebra of natural functions
No. Here's a counterexample: Let $f(r)$ satisfy the equation $f'' + 2 f^3 = 0$ with the initial conditions $f(0)=1$ and $f'(0)=0$. Then $f$ satisfies $(f')^2+f^4=1$ and is periodic with period
$$
L ...
- 108k
10
votes
Accepted
Diagonal analogue of symmetric functions
Yes, these have been studied before. They were studied by MacMahon under the name "symmetric functions of several systems of quantities." Nowadays they are usually called MacMahon symmetric ...
- 17k
2
votes
Stabilizers of the action of Levi on abelianization of nilpotent radical
This is an example of a "internal Chevalley module," which has been studied in a bunch of contexts. A good reference is ``On the structure of parabolic subgroups of algebraic groups'' by ...
- 2,516
2
votes
Stabilizers of the action of Levi on abelianization of nilpotent radical
I don't know the answer to your questions in this generality, but let me mention a rich class of examples. Let $P$ be the maximal parabolic corresponding to the highest root of $G$, and let $V = V_U$ ...
- 5,760
3
votes
Stabilizers of the action of Levi on abelianization of nilpotent radical
I am posting this as an answer since it is too long for a comment.
Consider the case that $G$ equals $\textbf{SL}(W)$ for a finite dimensional vector space $W$. Let $W=S\oplus Q$ be a direct sum ...
1
vote
Orbit spaces of n-tuples of square matrices under simultaneous conjugation
For $n=1$ your definition of $V_{\pi}$ implies that $G.V_{\pi}=M_p(\mathbb{C})$ for any partition $\pi$ of $p$, by the Jordan normal form.
For $n \geq 2$ one still has $G.V_{\pi} = G.V_{\sigma(\pi)}$ ...
- 4,267
Top 50 recent answers are included