# Questions tagged [symplectic-group]

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This question is a continuation of the question that I asked here: The principal congruence subgroup of the symplectic group over the integers Denote by $\Delta$ the group generated by $T=\{A\in \text{... 2 votes 0 answers 75 views ### Order of the symplectic group over$\mathbb{Z}/4\mathbb{Z}$[duplicate] Let$p$be a prime number and$q$some power of it. It is well-known that the order of the symplectic group$\text{Sp}_{2g}(\mathbb{F}_q)$over the finite field$\mathbb{F}_q$equals$q^{g^2}\prod_{i=... 1 vote
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### 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 ... 177 views

### Errata in N. A'Campo's "Tresses, monodromie et le groupe symplectique"

There are many small mistakes in this article. A great amount of them are concentrated in Lemma 2. The setup for this lemma is the following. Let $R$ be a commutative ring and $n=2g+1$ or $n=2g$ a ... 118 views

### When is the symplectic group over a commutative ring generated by its root subgroups and a maximal torus?

This is related to Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups. There I was told that in general, the symplectic group $\text{Sp}_{2n}(R)$ is not generated by its ... 181 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 ... 79 views

### The number of orbits of a two-point stabilizer of the symplectic group $Sp(2m,2)$

I am trying to figure out the number of orbits of a two-point stabilizer of the action of $Sp(2m,2)$ on its two orbits $\Omega^+$ and $\Omega^-$ as detailed in Dixon and Mortimer's "Permutation ...
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### Upper triangular similitude for symplectic matrices

It is known that given any matrix $M$ in $Sp(2,\mathbb{Z})$ with eigenvalue $+1$, we can find a real symplectic matrix $S$ such that $S^{-1}MS$ is upper triangular with diagonal entries equal to $+1$. ...
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### A trace formula for $\mathrm{GSp(4)}$

The Arthur trace formula and its variations provide general results for reductive groups, however to the extent of my knowledge only few specific instances of the formula have been really worked out ...
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### Do involutions always stabilize some transverse lagrangians?

Let $V$ be a vector space of dimension $2n\geq 4$ over a field $F$ of characteristic distinct from $2$. Assume that $V$ is equipped with a nondegenerate alternating form $b$. Let $Sp(V)$ denote the ...
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### Group of symplectic matrices over the field of complex numbers

According to Wikipedia, there are two different definitions of a "complex symplectic matrix" $M\in\mathbb{C}^{2n\times 2n}$. I am interested in the relation between the groups that follow from those ...
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Let $Sp(2n,{\mathbb R})$ be the symplectic group and $H_3(Sp(2n,{\mathbb R});{\mathbb Z})$ its 3rd group homology (i.e., for the group with the discrete topology). It is known that $$H_3(Sp(2n,{\... 4 votes 1 answer 128 views ### Solution of the Yang-Baxter equation associated to the U_q[osp(2n+2|2m)^{(2)}] Lie superalgebra I have a solution (a R matrix) of the Yang-Baxter equation, \begin{equation} R_{12}(x_{1})R_{13}(x_{1}x_{2})R_{23}(x_{2})=R_{23}(x_{2})R_{13}(x_{1}x_{2})R_{12}(x_{1}) \end{equation} that probably ... 3 votes 1 answer 187 views ### Generation of the symplectic by involutions Let G be a group. An involution is an element g\in G such that g^2=1. Let F be a field, V an F-vector space and b:V\times V \rightarrow F a nondegenerate alternating bilinear form. The ... 4 votes 1 answer 1k 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} ... 0 votes 1 answer 204 views ### Do tori in a symplectic group always have invariant maximal isotropic subspaces? \newcommand{\mbf}{\mathbf} Hi all, I've been thinking about the following question for a while now, and got a little stuck trying to solve it. Hopefully, someone here might be able to help. For ... 3 votes 0 answers 196 views ### Metaplectic groups over non-archimedean local fields of characteristic>2 Let K be a non-archimedean local field of characteristic >2. Consider the double cover metaplectic extension of symplectic groups p: Mp_{2n}(K)\rightarrow Sp_{2n}(... 2 votes 2 answers 308 views ### Symplectic form on the third symmetric power of a plane Let V a vector space of dimension 2 over a field k of characteristic different from 2 and 3. Let S^{3}V the third symmetric power of V. How to construct a symplectic form on S^{3}V ... 3 votes 1 answer 247 views ### Prescribed spherical representations, symplectic group Sp(n) An irreducible representation (\pi,V_\pi) of a compact group G is called spherical with respect to the pair (G,K), K is closed subgroup of G, if V_\pi has a non-zero vector invariant by K... 2 votes 0 answers 80 views ### explicit matrices for Weil (p^2 dimensional) representation of Sp(4,\mathbb{F}_p), p>3 I am looking for more-or-less explicit matrices for the p^2 dimensional Weil representation of Sp(4,\mathbb{F}_p), suitable for computer implementation. Ideally, I would like the images of the ... 6 votes 1 answer 913 views ### Structure of symplectic group over finite fields We are working over the finite field \mathbb{F}_{q} of odd prime characteristic p and of cardinality q some power of p. We recall the symplectic group Sp(4,\mathbb{F}_{q}) as the group of ... 7 votes 1 answer 234 views ### Exotic "non-linear" (but "almost linear") automorphisms of symplectic vector space Let V be a vector space over a field k equipped with a symplectic form \omega. Let f:V \rightarrow V be a bijective set map such that the following hold. For all v \in V and c \in k, we ... 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 ... 4 votes 1 answer 259 views ### In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator? Fix some g \geq 2, let \Gamma_g be the mapping class group of a genus g surface, and let \pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z}) be the projection. In Meyer, Werner Die Signatur von ... 13 votes 2 answers 456 views ### Bass's paper "Symplectic groups and modules", used in proof of the congruence subgroup property for Sp Let R be the ring of integers in a number field. While studying the congruence subgroup property for \text{Sp}_{2g}(R) in Bass, H.; Milnor, J.; Serre, J.-P. Solution of the congruence subgroup ... 1 vote 2 answers 878 views ### Is the metaplectic group not a matrix group - counterexample Is the statement below false? "The metaplectic group Mp2(R) is not a matrix group: it has no faithful finite-dimensional representations." Possible "counterexample": Sp(2n,R) is a subgroup of O(4n,C)... 0 votes 1 answer 424 views ### q_{S^*\omega}(X)=S^{\ast}q_{\omega}(X) ? Definition: Let (V,\Omega) be a symplectic vector space, we define \perp:\Lambda ^k(V^*)\to\Lambda ^{k-2}(V^{\ast}) by \perp(\omega)=i_{X_{\Omega}}(\omega) here if (e_1,e_2,...e_n,f_1,f_2,...... 5 votes 3 answers 923 views ### Finding generators of matrix subgroups I am particularly interested in Sp(2n,\mathbb{Z}), but I think an answer for a more general set of matrices would help. General question: Given a subgroup of a group of matrices, what tools or ... 7 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}, \...
Greetings to all ! Let me first confess that this question was mentionned to me by Bernard Dacorogna, who doesn't sail on MO. Let $A\in M_{2n}(k)$ be an alternate matrix. Say that $A$ is non-...