Questions tagged [symplectic-group]

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Congruence closure of principal congruence subgroup of the symplectic group over the integers

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{...
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2 votes
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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=...
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1 vote
1 answer
181 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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2 votes
0 answers
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 ...
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2 votes
1 answer
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 ...
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2 votes
1 answer
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 ...
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2 votes
0 answers
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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Maslov cycle for the Conley-Zehnder index - what are its regular points?

I'm looking at the definition of the Conley-Zehnder index, where it is important to look at the group $$\text{Sp}(2n)^* := \{ A \in \text{Sp}(2n) | \det (A-\text{Id}) \neq 0 \}$$and its complement $$\...
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3 votes
1 answer
223 views

Showing the positivity of the determinant of $\mathfrak{sp}(n)$ without making use of diagonalization

Let $\mathfrak{sp}(n)$ be the lie algebra of compact symplectic group $\mathrm{SP}(n)$, regarded as a compact form of $\mathfrak{sp}(2n,\mathbb{C})$, so we can talk about its (complex) determinant. ...
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4 votes
1 answer
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Is every $M\in \mathfrak{s}\mathfrak{p}_4(F)$ conjugate to an "upper triangular" matrix?

Let $F$ be a field and write $$\mathfrak{s}\mathfrak{p}_4(F)=\left\{\left(\begin{array}{cc} A & B \\ C & -A^T \\ \end{array}\right)\mid A,B,C\in M_2(F), B=B^T, C=C^T\right\}$$ for the ...
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5 votes
1 answer
193 views

Representations of $\operatorname{Sp}(2g,\mathbb{Z}_3)$

Let $V$ be a $2g$-dimensional vector space over $\mathbb{Z}_3 := \mathbb{Z}/3\mathbb{Z}$. First, $\operatorname{Sp}(2g,\mathbb{Z}_3)$ acts on $\Lambda^2(V)$, and this decomposition is reducible, as ...
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8 votes
1 answer
254 views

Subgroups of Sp(2g,Z) that map onto all Sp(2g,Z/m)

I stumbled into the following problem. I apologize for being a bit naive. For $g\geq 3$, consider the group $\mathrm{Sp}(2g,\mathbb{Z})$ of symplectic square matrices of order $2g$ with integral ...
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5 votes
0 answers
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Definition of modular Howe correspondence

Let $(G,G')$ be a pair of mutually centralized subgroups of a symplectic group $Sp_n(\mathbb{F}_q)$ (called a dual pair), and let $\omega_{G,G'}$ be the restriction of the Weil representation (with ...
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7 votes
1 answer
465 views

Bender-Knuth involutions for symplectic (King) tableaux

First let me recall the combinatorial theory of the characters of $\mathfrak{gl}_m$, a.k.a., Schur polynomials. For a partition $\lambda$, a semistandard Young tableaux of shape $\lambda$ is a filling ...
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2 votes
1 answer
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Lifting one parameter subgroup $e^{t K}$ to the universal cover of $\mathrm{Sp}(2N,\mathbb{R})$

I would like to lift an arbitrary one-parameter subgroup $e^{t K}$ with $K\in\mathfrak{sp}(2N,\mathbb{R})$ to the universal cover $\widetilde{\mathrm{Sp}}(2N,\mathbb{R})$ (or at least its two-fold ...
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6 votes
1 answer
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Paramodular newvectors and twists

In the book Local Newforms for GSp(4), Roberts and Schmidt have defined a theory of "new vectors" for smooth representations of $GSp_4$ over a nonarchimedean local field $F$ with trivial central ...
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1 vote
1 answer
150 views

On the number of involutions in some groups

How many involutions are there in $O_7(11)$ and $PSp_6(11)$ respectively? (Note that the sizes of the two groups mentioned here are the same.)
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4 votes
0 answers
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$\mathrm{Sp}_n(q)$-conjugacy classes in $\mathrm{GL}_{2n}(q)$

The symplectic group $\mathrm{Sp}_n(q)$ acts on $\mathrm{GL}_{2n}(q)$ by conjugation. All the literature I have found concerning the orbits of action of this kind is "Unipotent conjugacy classes in ...
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6 votes
2 answers
469 views

Conjugacy classes of symplectic group $\mathrm{Sp}(4,q)$

$\DeclareMathOperator\Sp{Sp}$I was reading the famous paper of Bhama Srinivasan "The characters of the finite symplectic group $\Sp(4,q)$". An AMS link for the paper is here. $\Sp(4,q)$ is ...
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1 vote
1 answer
171 views

Existence of symplectic basis

Let $R$ be a PID and $M$ a free, finite rank $R$-module with a perfect billinear form $\omega$ such that $\omega(v,v)=0$ for all $v \in M$. Does anyone know a reference for the fact that a symplectic ...
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4 votes
1 answer
180 views

Legendre's symbol in Schrödinger model for the Weil representation

I have a question concerning the Schrödinger model for the Weil representation over a finite field $\mathbb{F}_q$. The way to present the action of the Weil representation $\omega$ of $Sp(2n,\...
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20 votes
2 answers
1k views

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(...
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4 votes
1 answer
199 views

Are all these representations supercuspidal

Let $D$ a division quaternion algebra over a number field $F$, and consider $(V,q)$ be a $D$-hermitian space of $D$-dimension $2$, and introduce its group of isometries \begin{align*} \mathrm{GU}(V, q)...
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1 vote
1 answer
155 views

Invariant two form under symplectic group

Let $m,n\in\mathbb{N}$, $l$ be a prime number, let $J$ be the standard symplectic matrix $$J=\left[ \begin{array}[cc] \\0 & I_n \\ -I_n & 0\\ \end{array}\right]$$ Let $$\mathrm{Sp}(2n,\...
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5 votes
1 answer
262 views

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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2 votes
0 answers
124 views

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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2 votes
0 answers
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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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6 votes
4 answers
501 views

Zariski density of conjugates of $SL_2(\mathbb{Z})$ in $Sp_{2g}$

Let $Sp_2g$ be the symplectic group defined over $\mathbb{Q}$. Consider $SL_2(\mathbb{Z})$ as a subgroup of $Sp_{2g}(\mathbb{Z})$ (the embedding that I have in mind is $A\to \begin{pmatrix} A& 0\\...
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1 vote
0 answers
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Explicit matrices of representation of symplectic group

I would like to obtain explicit matrices for the representations of the Symplectic group $Sp_2(Z)$. For a pair of weights $(a, b)$ I know that the highest weight representations are contained in the ...
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14 votes
4 answers
965 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 $...
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6 votes
0 answers
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Homology of symplectic groups in the unstable range

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,{\...
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4 votes
1 answer
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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 ...
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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 ...
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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 $M\in F^{2n\times 2n}$ ...
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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 ...
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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}(...
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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 $S^{3}V$ ...
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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 $K$...
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2 votes
0 answers
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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 ...
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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 ...
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  • 85
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 ...
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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 $A$ ...
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  • 280
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 ...
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13 votes
2 answers
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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 ...
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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)...
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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,......
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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 ...
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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}, \...
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18 votes
4 answers
3k views

Alternate and symmetric matrices

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-...
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