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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0answers
36 views

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 $$\...
3
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1answer
203 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. ...
4
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1answer
87 views

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 ...
5
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1answer
175 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 ...
8
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1answer
192 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 ...
5
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38 views

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 ...
6
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1answer
399 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 ...
2
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1answer
105 views

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 ...
6
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1answer
98 views

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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1answer
145 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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85 views

$\mathrm{Sp}_n(q)$-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 ...
6
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2answers
408 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 ...
1
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1answer
164 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 ...
4
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1answer
161 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,\...
19
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2answers
921 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(...
4
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1answer
189 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)...
1
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1answer
153 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,\...
5
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1answer
247 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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0answers
122 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 ...
2
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0answers
558 views

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 ...
6
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4answers
472 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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0answers
150 views

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 ...
14
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4answers
898 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 $...
6
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279 views

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,{\...
4
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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 ...
3
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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 ...
3
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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 $M\in F^{2n\times 2n}$ ...
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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 ...
3
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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}(...
2
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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 $S^{3}V$ ...
3
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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 $K$...
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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 ...
5
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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 ...
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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 ...
2
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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 $A$ ...
4
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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 ...
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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 ...
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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)...
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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,......
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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 ...
6
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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}, \...
18
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4answers
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-...