# Questions tagged [classical-groups]

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59
questions

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### Involution centralizers in $\mathrm{PCSO}^{+}(8,3)$

According to Table 4.5.1 of [1], there should be 10 classes of involutions of type "p" and "e" in $\operatorname{Aut}(K)$ where $K=K_a=P\Omega^{+}(8,3)$.
And Table 4.5.1 also gives ...

1
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0
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52
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### Cardinal of finite orthogonal groups

Let $p \neq 2$ and let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$ and maximal ideal $\mathfrak{p}$.
By a quadratic space $V_{\mathcal{O}}$ of dimension $d$ over $\mathcal{O}$, I mean ...

2
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0
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141
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### What is the finite group $(\operatorname {PCO}^{\circ}_{2n})^{+}(q)$

In Table 22.1 on Page 193 of Malle & Testerman's book "Linear algebraic groups and finite groups of Lie type", the fixed point subgroup $G^F$ (where $F$ is a Steinberg endomorphism) of ...

1
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148
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### Bounds for the orders of second largest subgroups of $\mathrm{SL}_n(\mathbb F_q)$

$\DeclareMathOperator\SL{SL}$By Patton's thesis, except for a finite number of possibilities, the $(n-1, 1)$ parabolic subgroup, $P$ say, has the largest number of elements among all non-trivial ...

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1
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### Is there any (specially Algebraic Geometrical) exposition of Koike Terada's Young-diagrammatic methods for the representation theory paper?

I am talking about the paper by Koike, Kazuhiko and Terada, Itaru, Young-diagrammatic methods for the representation theory of the classical groups of type ($B_n$), ($C_n$), ($D_n$), J. Algebra 107, ...

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2
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423
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### Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime. It is easy to show that $\SL_2(3)$ can be embedded into $\SL_2(p)$.
Now, let $n$ be an integer larger than $2$.
Question: In which circumstances, $...

0
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0
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61
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### Involutions in $\operatorname {PSO}(4,K)$

In the algebraic group $G=\operatorname {PSO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there and what are the ...

0
votes

1
answer

100
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### An explicit matrix form in the symplectic group

In the algebraic group $G=\operatorname {PCSp}(2^{r},K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[...

0
votes

1
answer

85
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### An explicit matrix form

In the algebraic group $G=\operatorname {PCGO}(2m,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[
\...

1
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0
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105
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### Character table of $\mathrm{P\Gamma L}_2(q)$ with $q$ even

Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise ...

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154
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### Centraliser of a finite group

Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$.
We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...

1
vote

1
answer

272
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### Subgroups of $\operatorname{PGL}_n$

As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, $\operatorname{GO}_{2n}$ is a closed normal subgroup of the conformal orthogonal group $\operatorname{CO}_{2n}$. ...

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2
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210
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### is the embedding $\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ possible?

Is the following embedding possible?
$\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ where $S_{p^m-1}$ is a symmetric group and $p$ is prime. I see that when $p=3$ and $m=3$, the order of the former does ...

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0
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74
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### $C_G(E)= E \times{\rm PGL}_k(q)$

Let $r$ be an odd prime and $q$ a power of a prime $p$ where $r\neq p$.
If $r^m|n$ and $q\equiv1$ (mod $r$), then $r^{1+2m}.{\rm Sp}_{2m}(r)\le{\rm GL}_n(q)$ and $Center(r^{1+2m}.{\rm Sp}_{2m}(r))\le ...

1
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1
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139
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### Orbit sizes of $G=\operatorname{SO}^{+}_{2n}(2)$

Let $G=\operatorname{SO}^{+}_{2n}(2)$. I did some Magma computation and found there were $3$ orbits on the natural $G$-set when $n=2,3,4$. The orbit sizes are $1$-$9$-$6$, $1$-$35$-$28$, $1$-$135$-$...

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197
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### 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^{...

1
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1
answer

241
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### Kronecker product preserves the conjugacy relation?

Let $G =$ PGL$_{n}(\textbf{C})$ and $T$ be the image in $G$ of the subgroup of the invertible diagonal matrices of $\operatorname{GL}_{n}(\textbf{C})$. Let $A$ and $B$ be two elementary abelian $2$-...

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### Embedding (Kronecker product) preserves the structure?

In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix}
-I_{i} & 0\\
0 & I_{n-i}
\end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. ...

0
votes

1
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129
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### Intersection of identity components

Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...

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### elementary abelian subgroups with centralizers not connected

Let $G =$ PGL$_{8}(\textbf{C})$. Let $a, b, c, d$ be four representatives of conjugacy classes of involutions in $G$ where $$a = \begin{pmatrix}
-1 & 0\\
0 & I_{7}
\end{pmatrix}, b = \begin{...

4
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1
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126
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### Universal character ring for classical groups

The universal character ring for the general linear group is well understood but I want to ask about the universal character ring for the symplectic and orthogonal groups. For the general linear group,...

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136
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### Realization of a subgroup in a maximal subgroup of a classical group

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$In the finite group $G = \operatorname{PGL}_8(5)$, there is an elementary abelian $2$-subgroup $E$ of rank 5. $E = A_{1} \times A_{2} $ where $...

7
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330
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### $N_{G}(E)/C_{G}(E)$ is the Weyl group of $G$?

In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ denote the subgroup of elements of order dividing 2 in the diagonal maximal torus; it is generated by the images of the three ...

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0
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107
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### On the generation of linear groups

$\def\GL{\operatorname{GL}}\def\id{\mathrm{id}}$Let $V$ be a finite dimensional vector space over a (commutative) field $k$. If $f:V\to V$ is a linear map I'll write $r_f$ and $V^f$ for the rank of ...

2
votes

1
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135
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### An upper bound on the dimension of a subalgebra of $\mathfrak{so}(p,q)$ with non-trivial centre

Let $\mathfrak{so}(p,q)$ be the real definite/indefinite orthogonal Lie algebra, $p,q\ge0$, $p+q=n\in\mathbb{N}$, and $L\subset\mathfrak{so}(p,q)$ a Lie subalgebra with non-trivial centre, $\mathrm{Z}(...

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### Subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$

Is there a classification theorem for the subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$?
Apparently, there is an almost complete classification in ...

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0
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158
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### Minimal degrees of finite simple groups

The minimal projective degrees (minimal degree of an irreducible representation of a central extension) of the finite classical groups are (famously) given by Tiep and Zalesskii [1]. Is there a ...

1
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1
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159
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### Holomorphic map to Möbius group

$\DeclareMathOperator\PSL{PSL}$Let $U\subset\mathbb C^2$ be an open set, $f:U\to \PSL(2,\mathbb C)$ a holomorphic map. If the image of $f$ is contained in $\operatorname{PSU}(2,\mathbb C)$, I guess ...

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133
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### 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 ...

3
votes

1
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238
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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 ...

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0
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184
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### 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
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171
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### 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\...

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1
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### 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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### 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.
...

2
votes

2
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335
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### 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
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### 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, ...

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1
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### 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 ...

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2
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### 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
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489
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### 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

225
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### 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: ...

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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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### 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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1
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### $U(n)$-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 $X = \Gamma_{(2\lambda, \dots, 0)}$ ...

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### 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 ...

0
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154
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### 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
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396
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### 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
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769
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### 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
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182
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### 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-...

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268
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### 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 $G\ ?$

2
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### 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'...