# Questions tagged [symplectic-group]

The symplectic-group tag has no usage guidance.

54
questions

1
vote

0
answers

34
views

### Maximal subgroups of classical groups

Let G be a finite symplectic group Sp(4, GF(2^n)).
Is the smallest maximal subgroup of G known for all integers n?

3
votes

1
answer

174
views

### Normalisers and stabilisers in classical groups $\operatorname{PGL}_{4}$

In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{...

0
votes

0
answers

116
views

### Generators of the symplectic group

Let $n$ be an integer $\ge 1$. We define the $2n\times 2n$ matrix $\sigma$
with $n\times n$ blocks by
$$
\sigma=\begin{pmatrix}0&I_n
\\-I_n&0\end{pmatrix}.
$$
The symplectic group $Sp(n)$ is ...

4
votes

1
answer

125
views

### CW structure for $\mathrm{BSp}(n,\mathbb{C})$ and $\mathrm{BPSp}(n,\mathbb{C})$ in degrees $4i$

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\USp{USp}\DeclareMathOperator\BSp{BSp}\DeclareMathOperator\BUSp{BUSp}\DeclareMathOperator\BPSp{BPSp}$Let $\USp(n,\mathbb{C})...

5
votes

1
answer

190
views

### Motive associated to a cuspidal representation of $GSp_{4}$

In the paper by L.Clozel in this book (a French text), there is this conjecture (conjecture 4.5 p139)
Conjecture: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote ...

2
votes

0
answers

79
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

1
answer

261
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 ...

2
votes

0
answers

182
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 ...

2
votes

1
answer

147
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 ...

2
votes

1
answer

211
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 ...

2
votes

0
answers

110
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 ...

1
vote

0
answers

67
views

### 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 $+1$.
...

1
vote

0
answers

68
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
votes

1
answer

250
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
votes

1
answer

141
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
votes

1
answer

205
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
votes

1
answer

287
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
votes

0
answers

49
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 ...

7
votes

1
answer

572
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
votes

1
answer

122
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
votes

1
answer

114
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 ...

1
vote

1
answer

173
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.)

4
votes

0
answers

92
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
votes

2
answers

522
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
vote

1
answer

193
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
votes

1
answer

196
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,\...

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(...

4
votes

1
answer

205
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
vote

1
answer

161
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
votes

1
answer

274
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 ...

2
votes

0
answers

126
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
votes

0
answers

866
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
votes

4
answers

535
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\\...

1
vote

0
answers

174
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
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 $...

6
votes

0
answers

307
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
votes

1
answer

131
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

196
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

224
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

202
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

336
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

266
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

86
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

982
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

275
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

472
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

963
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)...