Questions tagged [symplectic-group]

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20 votes
2 answers
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The first unstable homotopy group of $Sp(n)$

Thanks to the fibrations \begin{align*} SO(n) \to SO(n+1) &\to S^n\\ SU(n) \to SU(n+1) &\to S^{2n+1}\\ Sp(n) \to Sp(n+1) &\to S^{4n+3} \end{align*} we know that \begin{align*} \pi_i(SO(...
Michael Albanese's user avatar
8 votes
1 answer
1k views

Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups

I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms. Consider the block matrices $$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \...
Maurizio Monge's user avatar
5 votes
1 answer
2k views

Symplectic group over integers and finite fields

For $H=\left( \begin{smallmatrix} 0 & I_n \\ -I_n & 0 \end{smallmatrix} \right)$ and a commutative ring $F$, the symplectic group $Sp(2n,F)$ is the set of all matrices $M\in F^{2n\times 2n}$ ...
Nicolas Malebranche's user avatar
15 votes
4 answers
1k views

Subgroups of $SL_2(\mathbb R)$ which contain $SL_2(\mathbb Z)$ as a finite index subgroup

Let $G\subset \mathrm{SL}_2(\mathbb R)$ be a subgroup such that $\mathrm{SL}_2(\mathbb Z)\subset G$. What are the possible groups such that $\mathrm{SL}_2(\mathbb Z)\subset G$ is of finite index? Is $...
Honing's user avatar
  • 195
3 votes
1 answer
259 views

Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups

This is a question about the answer in this other post: Symplectic group over integers and finite fields. In general, for any ring $R$, the symplectic group $\text{Sp}(2n,R)$ is generated by its root ...
user avatar
2 votes
1 answer
1k views

Symplectic block-diagonalization of a complex symmetric matrix

This is a follow-up question to the one asked here: Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...
Fabian's user avatar
  • 280
1 vote
1 answer
335 views

The principal congruence subgroup of the symplectic group over the integers

Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are ...
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