# Questions tagged [classical-groups]

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31
questions

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**1**answer

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### On $(2,3)$-generation of finite simple classical groups

A group $G$ is called $(a,b)$-generated if $G=\langle x,y\rangle$ for some $x,y\in G$ with $|x|=a$ and $|y|=b$.
I know some of the histories on this problem. For example, in this early paper in 1996 ...

**1**

vote

**0**answers

89 views

### Cohomology ring of special linear group over finite fields

I am trying to find about the cohomology ring $H^*(SL_n(\mathbb{F}_q),\mathbb{Z}/2\mathbb{Z})$ where $q$ is odd. For $n=2$, an explicit description is given. But for $n>2$, I didn't come across a ...

**2**

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**2**answers

131 views

### Existence (or the number) of generating triple of involutions of $\operatorname{PGL}_2(p)$ with some conditions

Let $G=\operatorname{PGL}_2(p)$, where $p\ge 5$ is a prime. Is there a generating triple of involutions $(x,y,z)$ of $G$ such that $|xy|=p$, $|xz|=p+1$ and $|yz|=p-1$? That means, $\langle x,y,z\...

**3**

votes

**1**answer

94 views

### Is the Singer cycle preserved by field automorphisms and graph automorphisms?

Let $T=\operatorname{PSL}_n(q)$ with $n$ a prime number. Then the $\mathscr{C}_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}_{\frac{q^n-1}{(q-1)(n,q-1)}}{:...

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104 views

### Normaliser of the field automorphism of $\operatorname{GL}_n(p^f)$ in $\operatorname{GL}_{fn}(p)$

I have asked the same question on MSE several days ago here but there was no any useful comments. So I try to ask here.
Let $G=\operatorname{GL}_{fn}(p)$, where $p$ is a prime and $fn>2$, and $H$ ...

**10**

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256 views

### Fake degrees: why coinvariant algebra and classical groups over finite fields?

Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated.
...

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263 views

### Is the size of a conjugacy class in a finite classical group a polynomial?

Suppose $G$ is a classical matrix group over a finite field of order $q$.
If $C$ is a conjugacy class in $G$ , is $|C|$ a polynomial in $q$?
This question is supported by the fact that whenever I ...

**3**

votes

**1**answer

101 views

### Splitting of regular semisimple conjugacy classes in $SL_{n}(q)$

I have the following question: Consider the following two finite groups: $GL_{n}(q)$ and $SL_{n}(q)$. What I am trying to understand is the regular semisimple conjugacy classes of $SL_{n}(q)$. Now, ...

**11**

votes

**1**answer

166 views

### Can elements in the orthogonal group of a non-split Azumaya algebra with an orthogonal involution have reduced norm -1?

Let $R$ be a connected (commutative) ring with $2\in R^\times$.
Let $A$ be an Azumaya algebra over $R$ and let $\sigma:A\to A$ be an orthogonal involution. (This means that there is a faithfully flat ...

**22**

votes

**2**answers

1k views

### Why is the catalecticant invariant under coordinate changes?

Let $\mathbf{k}$ be a commutative $\mathbb{Q}$-algebra. (We could play the
same game over any commutative ring $\mathbf{k}$, but this would be a bit more
technical, so let me avoid it.)
Fix a ...

**10**

votes

**1**answer

407 views

### The double cover of $[W(E_7),W(E_7)] \cong Sp_6(\mathbb F_2)$ as a Galois group over $\mathbb Q$

I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$:
Is the double cover of $Sp_6(...

**2**

votes

**1**answer

164 views

### unipotent class in classical lie algebra bala-carter

For a nilpotent class in a semi-simple Lie algebra that is well determined by its weight Dynkin diagram, according to Bala-Carter, how can I determine the associated nilpotent matrix? For example: ...

**2**

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**0**answers

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 ...

**3**

votes

**1**answer

185 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 ...

**6**

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**1**answer

191 views

### $U(n)$-submodules of ${\rm SO}(2n)$-modules

Let $\Gamma_{(\lambda_1, \dots, \lambda_{n})}$ denote an irreducible $SO(2n)$-module with highest weight $(\lambda_1, \dots, \lambda_n)$ and let more specifically $X = \Gamma_{(2\lambda, \dots, 0)}$ ...

**6**

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**2**answers

1k views

### Describing Levi factors and unipotent radicals of parabolic subgroups in classical groups

I asked this question before at Math.SE (link) but got no answer.
Let $G$ be an algebraic group over an algebraically closed field $k$ of characteristic $p \geq 0$. Then any parabolic subgroup $P$ of ...

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**0**answers

146 views

### Buildings for Affine groups

Let $G$ denote one of the classical groups over a finite field. Is there a natural way to associate a building to the affine group $V\rtimes G$, and an analog of the Solomon-Tits theorem?

**2**

votes

**1**answer

305 views

### Compact form of symplectic groups defined over the rationals

I am a bit confused regarding the possible constructions/realizations of symplectic groups. Basically I am looking for the following:
A linear algebraic group $\mathbb{G}$ defined over $\mathbb{Q}$ ...

**6**

votes

**2**answers

557 views

### Extensions of $SL(2,\mathbb{F}_q)$

Let $n = q(q^2-1)$ (the order of $SL(2,\mathbb{F}_q)$ I think). How many groups of order $2n$ contain $SL(2,\mathbb{F}_q)$ as a (necessarily normal) subgroup? Is this number known exactly? Seems ...

**2**

votes

**1**answer

172 views

### Stabilizer of a nonsingular vector in a quadratic space (char (k)=2)

suppose that $k$ is a finite field of characteristic 2 and $(V,q)$ a quadratic space, i.e., $V$ is a $k$-vector space and $q:V\to k$ quadratic form. Suppose that $\dim(V)\geq 4$ and that $q$ is non-...

**2**

votes

**1**answer

239 views

### question about projective special unitary group

Let $q$ be odd, $G=PSU_n(q)$ (Projective Special Unitary group) and $H=PSU_{n-1}(q)$. Is it always true that $H$ is a subgroup of $G\ ?$

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votes

**1**answer

219 views

### Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's “Classical Groups”

First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but didn'...

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**0**answers

247 views

### multidimensional rotation terminology

Given an element $g$ of the orthogonal group $O(n)$, is there a name for the subspace of $R^n$ that's fixed by $g$, and a name for the orthogonal complement of this space? (The latter is what I ...

**18**

votes

**1**answer

960 views

### Existance of certain almost invariant functions related to amenability and piece-wise transformations

We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...

**14**

votes

**3**answers

3k views

### Connectedness of the linear algebraic group SO_n

I apologize in advance if my question is too elementary for MO.
It is a well known fact that the linear algebraic group $G = \mathsf{SO}_n$ is connected, and there exist a few different proofs of ...

**1**

vote

**0**answers

499 views

### Totally singular subspaces in orthogonal vector spaces

This is for all that are interested in classical groups and their representations.
We are investigating the following situation:
Let $V$ be $d$-dimensional $k$-vector space (where $k$ is a finite ...

**6**

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}, \...

**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-...

**14**

votes

**2**answers

2k views

### What is the subgroup generated by involutions?

I was recently taking some notes on the Cartan-Dieudonné theorem: if $(V,q)$ is a nondegenerate quadratic space of finite dimension $n$ over a field of characteristic not $2$, then every element of ...

**44**

votes

**7**answers

5k views

### Does $SL_3(R)$ embed in $SL_2(R)$?

Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$?
$SL_{3}(\mathbb{Z})$ is not an amalgam, and has the wrong number of order $2$ elements to be a subgroup ...

**4**

votes

**2**answers

2k views

### When does an irreducible representation remain irreducible after restriction to a semi-simple subgroup?

I suppose this question is probably elementary for experts, but I'd like to present my arguments, about which I have some doubts, and see if they are correct, or if corrections and improvements are ...