0
votes
0answers
63 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
2
votes
1answer
95 views

Inclusion $O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $

How to show, that an inclusion of homogenious spaces $$O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $$ is homotopy equivalence? The big space is the space of complex structures on ...
2
votes
2answers
275 views

why symplectic form should be closed when we work on a manifold

For defining the symplectic space $(V, \omega)$ where $V$ is a vector space, it doesn't necessary to add the condition $d\omega=0$. But, when we work on a manifold instead of vector space, then we ...
2
votes
0answers
45 views

Trivialization of a path of tamed almost complex structures

I am wondering if the following result is true: Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by ...
2
votes
1answer
247 views

Any intuitive examples of symplectic vector space?

Recently I come into symplectic vector space and its properties in my linear algebra class. However, this interesting thing is so different from the usual inner-product spaces I've met before, and I ...
1
vote
1answer
49 views

Given an integral symplectic matrix and a primitive vector, is their product also primitive?

Given a matrix $A \in Sp(k,\mathbb{Z})$, and a column k-vector $g$ that is primitive ( $g \neq kr$ for any integer k and any column k-vector $r$), why does it follow that $Ag$ is also primitive? Can ...
1
vote
1answer
101 views

Symplectic basis $(A_i,B_i)$ such that $S= $ span$(A_1,…,A_k)$ for some $k$ when $S$ is isotropic

Let $(V,\omega)$ be a symplectic vector space of dimension $2n$. How can I show that for an isotropic subspace $S \subset V$, there exists a symplectic basis $(A_i,B_i)$ for $V$ such that $S= $ ...
2
votes
1answer
149 views

Symplectic basis $(A_i,B_i)$ such that $S= $ span$(A_1,B_1,…,A_k,B_k)$ for some $k$ when $S$ is symplectic

Let $(V,\omega)$ be a symplectic vector space of dimension $2n$. How can I show that for a symplectic subspace $S \subset V$, there exists a symplectic basis $(A_i,B_i)$ such that $S= $ ...
3
votes
0answers
266 views

basis free volume form for a symplectic vector space

It's easy to show, using a symplectic basis, that if $\omega$ is a symplectic form on a $2n$-dimensional vector space $V$, then $\omega^n \neq 0$. I'd like to be able to prove it without choosing a ...