# Tagged Questions

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*} ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $S=$ ...
### 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 $S=$ ...
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 ...