All Questions
7 questions
4
votes
1
answer
252
views
Flag manifolds as homogeneous Kahler manifolds
In this question it is asked if every flag manifold can be given the structure of a Kähler manifold. In the first answer it is written
Flag manifolds exhaust all compact homogeneous Kähler ...
1
vote
1
answer
408
views
De Rham cohomology of homogeneous spaces
Take a homogeneous space $K/L$, where $K$ and $L$ are compact lie groups. Denoting by $\Omega^*(K/L)$ its de Rham complex, which is a homogeneous vector bundle over $K/L$, and hence has a ...
3
votes
0
answers
72
views
Homogeneous space for intersection of subgroups
Suppose we have a Lie group $G$ and two subgroups $P_1$ and $P_2$. We can then study the homogeneous spaces $M_1=G/P_1$ and $M_2=G/P_2$, and bundles on these spaces associated to representations of $...
- 1,488
5
votes
1
answer
902
views
Explicit description of the Lagrangian Grassmannian as a homogeneous space
Looking at this and this question about the Lagrangian Grassmannian, and its linked Wikipedia description as the quotient of $Sp(N)$ by the unitary group $U(n)$, I wondering what is the explicit ...
- 459
2
votes
0
answers
161
views
Intermediate quotient for a Hermitian Symmetric Spaces of $Sp(n)$
We know that $U(N)$ can be embedded into $SU(n+1)$ and that the quotient is isomorphic to complex projective space:
$$
SU(n+1)/U(n) \simeq {\mathbb CP}^{n}.
$$
We can split this process into two ...
- 167
1
vote
1
answer
930
views
Canonical class of partial flag variety
Let $F(d_1,d_2,\ldots,d_k;n )$ be the variety of all flags $\mathbb A^{d_1} \subset\mathbb A^{d_2}\subset\ldots\subset \mathbb A^{d_k}\subset \mathbb A^{n}$. This variety has the natural maps to ...
- 600
8
votes
1
answer
566
views
Quotienting $SU(3)$ by $U(1)$?
As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...
- 155