# Questions tagged [symplectic-group]

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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,{\... 1answer 122 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 ... 1answer 183 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 ... 1answer 701 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} ... 1answer 165 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 ... 0answers 184 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}(... 2answers 292 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 ... 1answer 226 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... 0answers 78 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 ... 1answer 790 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 ... 1answer 225 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 ... 1answer 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 ... 1answer 238 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 ... 2answers 430 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 ... 2answers 822 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)... 1answer 423 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,...... 3answers 806 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 ... 1answer 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-...