13 votes
Accepted

Maximum symmetry metric on $ \mathbb{C}P^n $

There's an easy counterexample to your guess: Let $M^6 = \mathrm{SU}(3)/\mathbb{T}^2$, where $\mathbb{T}^2\subset\mathrm{SU}(3)$ is the maximal torus (for example, the diagonal subgroup). In that ...
5 votes

Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups

Definitely not a full answer (and rather late to the party) but I am interested in the $E_7$ case myself so I have been doing a little computing of the dimensions of these flags that I thought was ...
  • 747
5 votes
Accepted

Injectivity of the cohomology map induced by some projection map

Ok, I will follow Fernando's advice and post an answer. I learned the computation below from the beginning of Pin(2)-equivariant Seiberg--Witten Floer homology and the triangulation conjecture. The ...
  • 8,728
5 votes
Accepted

Question about maximal compact subgroups of Lie groups

$\DeclareMathOperator\Lie{Lie}\newcommand\g{\mathfrak g}\newcommand\C{{\mathbb C}}$$\g = \Lie(G)$ is maximal among subalgebras of $\g_\C = \Lie(G_\C)$ on which the Killing form is negative definite, ...
  • 9,057
5 votes

Maximum symmetry metric on $ \mathbb{C}P^n $

I just wanted to add two points: A bi-invariant metric on a compact Lie group $G$ does not always induced the maximum symmetric metric on $G/H$. The most familiar examples are spheres: $S^{2n+1} = ...
  • 6,011
4 votes
Accepted

Generalization of $G/T \simeq G_\mathbb{C}/B$

No. Take $G = SU(2)$, $G_{\mathbb C} = SL_2(\mathbb C)$, $G/B$ the complex projective line alias the sphere, $H$ the diagonal $U(1)$, $x$ any point other than the two fixed points of $H$, so that the ...
  • 122k
4 votes

Injectivity of the cohomology map induced by some projection map

OK, just noticed mme's comment, which considers the very same counterexample below with a clean one-line argument. I'm leaving this here in case someone finds something of any value, but I think mme's ...
2 votes
Accepted

Free $S^1$-action on compact homogeneous spaces

Here is a counterexample where $K$ is connected. Let $W = SU_3/SO_3$ be the Wu manifold. This 5-dimensional manifold is simply-connected and has $H_2(W;\Bbb Z) = \Bbb Z/2$, and in particular, $\pi_2(W)...
  • 8,728
2 votes
Accepted

Question about coadjoint orbits of compact connected Lie groups

$\DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\ad}{ad} \DeclareMathOperator{\Lie}{Lie} \newcommand{\g}{{\mathfrak g}} \newcommand{\z}{{\mathfrak z}} \newcommand{\s}{{\mathfrak s}} \newcommand{\O}...
2 votes

Is it possible to average a riemannian metric over an action and preserve curvature bounds?

As it was mentioned by Igor Belegradek, the curvature gets destroyed by averaging. Assume there is a modified the averaging process that preserves positive curvature. Then you would prove Hopf ...
1 vote

Why is $\mathrm{SO}(4)$ not a simple Lie group?

The question is somewhat open-ended — more about beauty than truth. So I will offer two rather different narratives. One way a group $G$ can fail to be simple is that it has a representation which is ...
1 vote

Counting adjoints in the symmetric or antisymmetric square of a Lie group representation

I don't have a proof but I noticed some things while having a quick look for some counterexamples that I thought were worth sharing. First things first what you are looking for is not special to the ...
  • 747

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