15
votes
Accepted
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$ as $\operatorname{GL}(n,\mathbb{C})$-modules
This is a problem about plethysm. Example 9 from Macdonald's book, in the section on plethysm, gives the formulas
$$ e_2 \circ e_k = \sum_{j \text{ odd}} s_{(k+j,k-j)'}$$
and
$$ h_2 \circ e_k = \sum_{...
- 40.2k
7
votes
Accepted
Lie subalgebra annihilated by all derivations
A class of counterexamples is given by the so-called characteristically nilpotent Lie algebras. For a Lie algebra $\mathfrak{g}$, consider the central descending chain defined recursively by
$$\...
- 5,878
5
votes
Non-semisimple Lie groups and Higgs bundles
One can replace $\mathfrak m$ by $\mathfrak g/\mathfrak h$ where $\mathfrak g$ is the Lie algebra of $G$ and $\mathfrak h$ is the Lie algebra of $H$. We clearly have $[\mathfrak h,\mathfrak h ] \...
- 148k
5
votes
Accepted
Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$
$\def\Spec{\text{Spec}}\def\GG{\mathbb{G}}\def\ZZ{\mathbb{Z}}$Yes, there are other Hopf structures. First I'll give a non-local (but connected) example, and then I'll modify it to be local.
...
- 156k
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
2
votes
Non-semisimple Lie groups and Higgs bundles
There is a (related but not quite the same) construction which is valid for any Lie group $G$ and any closed (hence Lie) subgroup $H\subset G$ over any smooth base manifold $X$ which may be helpful.
...
- 7,817
2
votes
Accepted
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I believe that the answer to your question is yes and it follows from Proposition 8.1 in our paper The Mirković–Vilonen basis and Duistermaat–Heckman measures (with Baumann and Knutson). The notation ...
- 4,612
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