4
votes
1answer
222 views

The Maslov triple product is alternating in its entries

Let $(V,\omega)$ be a $2g$-dimensional symplectic vector space. I'm trying to understand the Maslov triple product. I know that it can be defined in a variety of ways, but for the applications I'm ...
3
votes
1answer
451 views

Symplectic block-diagonalization of a real symmetric Hamiltonian matrix

Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal? Being ...
1
vote
1answer
163 views

if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic?

Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form $q: \Pi \oplus \Pi \rightarrow R $ $\big( \alpha ,\beta \big) \longrightarrow \alpha ...
1
vote
0answers
125 views

Comparing the volume of a rational lagrangian under a linear symplectomorphism.

Let's fix the standard symplectic structure $(\mathbb{R}^{2g}, \omega, J)$. A (marked) symplectic lattice then has the form $A\mathbb{Z}^{2g}$ for $A \in Sp_{2g}\mathbb{R}$. We say a vector subspace ...
3
votes
4answers
479 views

Polar decomposition for quaternionic matrices?

A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, ...
0
votes
1answer
236 views

intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$

Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. ...
1
vote
0answers
193 views

(Non-)Surjectivity of the Maslov index

Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over ...
0
votes
1answer
272 views

Finding the $J$ for a symplectic vector space

I found something strange when I was working on some other problems. I thought the triple intersection description of the unitary group said that any two of $(g, \omega, J)$ determines the third ...
2
votes
1answer
450 views

How do you construct a symplectic basis on a lattice?

Is this possible to do constructively? The only sources that I have for the possibility of this construction is an exercise in Lang's Algebra (on p. 598, I believe) which states that one can be ...