Questions tagged [classical-groups]

The classical groups are the general and special linear groups over the reals, the complex numbers and the quaternions, together with the automorphism groups of certain non-degenerate forms. These are symmetric or skew-symmetric bilinear forms over the reals or the complex numbers, hermitian forms over the complex numbers or the quaternions and skew-hermitian forms over the quaternions.

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Involutions in PCO

In the algebraic group $G=\operatorname {PCO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there in $G \setminus \...
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Fundmental groups of matrix groups and covering relation between them [closed]

I am self studying algebraic topology, the fundamental groups and covering spaces and homologous groups. I have read the topological properties of classical matrix groups like connectedness, path ...
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maximal subgroup of the general linear group

Maybe I'm being silly. As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, is the conformal orthogonal group $CO_{2n}$ a maximal subgroup of $GL_{2n}$(or $CO_{2n+1}$ ...
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An explicit isomorphism between $\mathbb{R}^+ \times {\rm Spin}^c(3,1)$ and ${\rm GL}^+(2,\mathbb{R})\times {\rm GL}^+(2,\mathbb{R})$? [closed]

I am interested in the following isomorphism $$ \begin{align} \mathbb{R}^+\times {\rm Spin}^c(3,1)& \cong \mathbb{R}^+\times {\rm Spin}(3,1) \times {\rm U}(1) \tag{1}\\ &\cong \mathbb{C}^+\...
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Choice of generators to make the centralisers connected

In $G=\operatorname{PGL}_{2n}(\textbf{C})$, WLG, we assume all the toral elementary abelian 2-subgroups in discussion are in $T$, the image in $G$ of the group of diagonal matrices in $\operatorname{...
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1 vote
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Transitive and Imprimitive group action with certain orbit lengths

I am reading the book of D.E.Taylor the Geometry of Classical Groups and it is asked that: Let $G$ act transitively on a set $\Omega$ of size $n$ and that $Stab_{G}(\omega)$ has orbits of lengths $1$, ...
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Stabiliser group [closed]

Let $G=\operatorname{Sp}_{2r}(2)$. There are two orbits of $G$ on the natural $G$-set, one having the identity, the other having all the remaining elements. What is the subgroup of $G$ that stabilizes ...
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Conjugate subgroups by permutation matrices

Is the following statement true or false? In $G=\operatorname{GL}_{n}(\textbf{C})$, two elementary abelian $2$-subgroups $X$ and $Y$ of $G$ whose generators are all diagonal matrices are conjugate if ...
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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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What is the Order Of Symplectic Group(4,2) and Symplectic Group(4,3) from the Classical groups of Atlas?

I have been working on Symplectic group of classical groups. I am trying to find Sylow-2 Subgroups of Symplectic Group(4,2) and Symplectic Grouop(4,3) through GAP, I am facing a problem regarding ...
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$N/C$ contains $\operatorname{GL}_{2}(2)$

In $G = \operatorname{PGL}_{4}(\mathbb{C})$, a subgroup $E = <a,b,c,d>$ where $$a =\Delta (\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix})_{2}, b =\Delta (\begin{pmatrix} 0 & 1\\ 1 &...
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intersection of point stabilisers is trivial

Let $G=\operatorname{GL}_{n}(2)$. Let $v_{i}$ be the basis elements of the natural module of $G$. I observed by computing with Magma that the intersection of all Stabiliser($G, v_{i}$) is trivial for ...
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Linear group contained in symplectic group

Do we have SL$(n,2) \leqslant$ Sp$(2n, 2)$? And better yet, $\operatorname{SL}(n,q) \leqslant$ Sp$(2n, q)$ or $\operatorname{GL}(n,q) \leqslant$ Sp$(2n, q)$? I checked using Magma for small degrees ...
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Matrix group inclusion

It is true that $Sp_{2}(2)$ is a subgroup of the matrix group $\begin{pmatrix} Sp_2(2) & 0\\ *_{1\times 2} & 1\\ \end{pmatrix}$ where $*_{1\times2}$ denotes a $1\times 2$ matrix with ...
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Construction of $SL(2,5)^4$ in a maximal subgroup in Magma

There is a Class $2$ maximal subgroup $SL(2,5)^{4}.4^{3}.S_{4}$ (denoted by $MM_2$ in the following code) in $PGL(8,5)$. I am trying to locate the $SL(2,5)^4$ part in Magma. But after I constructed ...
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construct the normaliser of a subgroup and then construct the subgroup

There is a maximal nontoral (not contained in a conjugate of a fixed maximal torus) elementary abelian $2$-subgroup of rank 6 in $G = PGL(8,7)$. I denote this group by $A$. Its normaliser $N_{G}(A)$ ...
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split maximal torus construction

In PGL(n,q) there is a split maximal torus T of order $(q-1)^{n-1}$. How to construct this in Magma? Let's use the example of $PGL(4,11)$. I took a detour to construct it: ...
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inclusion among finite symplectic groups

This might be a silly question, but do we have the following? $Sp(2,q) < Sp(4,q) < Sp(6,q) < Sp(8,q) <....$ I checked the list of maximal subgroups of $Sp(n,q)$ for $n = 4,6,8,10,12$ and ...
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representatives of conjugacy classes of involutions

I'm aware that the representatives of conjugacy classes of involutions of $G = PGL(4,\mathbb{C})$ which have a conjugate in a fixed maximal torus of $G$ are $\begin{bmatrix} -1 & 0 & 0\\ 0 &...
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6 votes
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Explicit matrices $h_\alpha$ that correspond to the long roots $\alpha$ in a classical compact simple Lie algebra over the reals

While it is easy to find many sources that give expressions for the (co)roots of an abstract root system, it is less easy to find a reference that gives explicit matrices that are the "coroots&...
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Automorphism group of $\text{GL}(n,\mathbb{F}_q)$

In the book On the Automorphisms of the Classical Groups, J. Dieudonné the author describes all the automorphisms of $\text{GL}(n,\mathbb{F}_q)$, but I couldn't find a description of the automorphism ...
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Uniqueness of maximal compact group

I'm studying the maximal compact subgroups of $PSL(2,\mathbb C)$. According to Wikipedia, "The Cartan-Iwasawa-Malcev theorem asserts that every connected Lie group (and indeed every connected ...
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Is $\mathbb{C}^{n}$ an irreducible representation of $O(n, \mathbb{C})$?

I am trying to prove that $\mathbb{C}^{n}$ is an irreducible representation of $O(n, \mathbb{C}) = \left\{ A \in GL(n) | A^{t}A = Id \right\}$. I presume the action is defined in the standard way, i.e....
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Classification of Continuous Group Homs from Circle to $SL_2(\mathbb{R})$

Problem. Classify all continuous group homomorphisms $f:\mathbb{S}^1\to SL_2(\mathbb{R})$. Attempt 1. My initial thought was to look at the induced Lie algebra map $df:\mathbb{R}\to \mathfrak{sl}_2$. ...
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Generation of $SU(n,q^2)$ by subgroups isomorphic to $SU(2,q^2)$

Throughout, let $q$ be an odd prime power. Let $GF(q^2)$ be the field with $q^2$ elements. My question concerns the generation of the special unitary group $SU(n,q^2)$, where $n \ge 2$, by certain ...
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Does $SL_2(K) \simeq SL_2(L)$ imply $K\simeq L$?

Let $K$ and $L$ be two fields. Assume characteristics are not 2. I can show in a quite elementary way that if the statement $SL_2(K) \simeq SL_2(L) \implies K \simeq L$ holds, then for $n \geq 2$, the ...
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Results on Maximal subgroups of $\text{GL}(n,q)$

Consider, $\text{GL}(n,q)$, which is the group of $n\times n$ invertible matrices with entries from the finite fields $\mathbb{F}_q$, with $q$ elements.(We know $q$ is a prime-power). My question is ...
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$K/k$ Galois with group $\mathfrak{g}$, $V_{K} \cong V_{k} \otimes K$. Then, $V_{k}$ consists of elements of $V_{K}$ invariant under $\mathfrak{g}$.

I have been asked to read "Andre A. Weil. Algebras with involutions and the classical groups. J. Indian Math. Soc.(N.S.) 24 (1960), 589–623" as part of a project, and I am encountering some ...
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1 answer
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Maximal subgroups of ${\rm PSL}(2,8)$ [duplicate]

I’m reading a note on the subgroup structure of classical subgroups. According to Corollary 2.2 (d), since $8=2^3$ is a prime power of even number $2$, ${\rm PSL}(2,8)$ has a maximal subgroup ${\rm ...
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1 answer
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Is $GL_{n}(\Bbb{C})$ isomorphism to a subspace of $GL_{2n}(\Bbb{R})$

A problem in the Algebra by Artin. Is $GL_n(\Bbb{C})$ isomorphism to a subspace of $GL_{2n}(\Bbb{R})$ I think there is an isomorphism. Because I know that when $n=1$, $$\{A\mid A=\left(\begin{matrix}...
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Is a certain square root of $-1$ an element of $\Omega_{4l}^+(q)$?

Let $F$ be a finite field of order a power $p^r$ of some odd prime $p$ for some positive integer $r$, and $c$ a the generator of $F^{\times}$ , is $$\begin{pmatrix} 0 & A_{2l\times 2l}\\ B_{2l\...
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0 answers
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Confusion over spin representation and coordinate ring of maximal orthogonal Grassmannian

I'm working over $\mathbb{C}$ here. If we let $G=\mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the maximal parabolic corresponding to the $1$st node in the Type $B_n$ Dynkin diagram (...
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4 votes
0 answers
450 views

Understanding Conjugacy classes of the Unitary group over finite fields

Consider the General linear group $GL(n,q)$ over the finite field $\mathbb{F}_q$ of $q$ elements. The unitary group $U(n,q)$ is described as a subgroup of $GL(n,q^2)$ which is the set of linear maps ...
Riju's user avatar
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2 votes
0 answers
42 views

Intersection of isometric subspaces in a space over a finite field

Suppose $V$ is a symplectic, unitary, or orthogonal space over a finite field $\mathbb{F}_q$ of characteristic $p$. Suppose $H\subseteq U\subseteq W$ is a chain of subspaces of $V$, where $\dim U=\...
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1 answer
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Is $PGL(n,\Bbb{R}) \cong SL(n+1, \Bbb{R})$ for even n?

Is the following claim correct? Claim: If n is even, $PGL(n,\Bbb{R}) \cong SL(n+1, \Bbb{R})$. Proof: Recall $PGL(n,\Bbb{R}) \cong GL(n+1, \Bbb{R})/Z$, where $Z = \{M | M=\alpha I, \alpha \in \Bbb{...
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2 votes
1 answer
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Transitivity of $\Omega$ in subspace actions

Let $G=O(V, Q)$ be a finite orthogonal group acting naturally on a space $V\cong \mathbb{F}_q^n$ equipped with a quadratic form $Q$. Assume $n=\dim V\geq c$ for some large enough constant $c$ in order ...
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3 votes
1 answer
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Reference request: structure of stabilisers of totally isotropic subspaces in orthogonal (and unitary) groups

I am looking for a book or paper which covers the structure of stabilisers in $GO(n,F)$, $SO(n,F)$ (or maybe in $\Omega(n,F)$) of totally isotropic subspaces of dimension $k$. Can you please suggest ...
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2 votes
1 answer
93 views

Sylow subgroup of a subgroup 5

I am aiming for looking for all the Sylow subgroups of classical groups, which gives me a seemingly elementary question: what can we say about Sylow subgroups of a subgroup if we know all Sylow ...
Marcos G Neil's user avatar
1 vote
1 answer
116 views

Does $\text{SU}(1,1)$ act transitively on $\mathbb S^1=\{z\in\mathbb C\mid |z|=1\}$?

Let $\text{SU}(1,1)=\left\{\left[ \begin{array}{ccc} \alpha & \beta \\\overline\beta & \overline\alpha \end{array} \right]\mid \alpha,\beta\in \mathbb C,|\alpha|^2-|\beta|^2=1\right\}$ and $\...
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2 votes
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Discovering $\pi_1(SO(3))$

One can use the double covering of $\text {SO}(3)$ by $\text {SU}(2)$ to compute the fundamental group of $\text {SO}(3)$. I’d like to travel in the opposite direction, using the definition of $\...
Jyotirmoy Bhattacharya's user avatar
2 votes
0 answers
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How to determine if two algebra automorphisms of $M_n(\mathbb{R})$ are similar?

Given two algebra automorphisms $\phi, \psi: M_n(\mathbb{R}) \to M_n(\mathbb{R})$ of the real algebra $M_n(\mathbb{R})$, we say they are similar if there exists an algebra automorphism $\alpha: M_n(\...
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1 answer
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Hall subgroups of $\mathrm{PSL}$

The following is an exercise in Peter Cameron's notes on classical groups. Exercise 2.10 (a) Show that $\mathrm{PSL}(2,5)$ fails to have a Hall subgroup of some admissible order. (b) Show that $\...
Groups's user avatar
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1 vote
1 answer
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Notions of Fundamental Groups for semisimple algebraic groups

Let $G$ be a connected, semisimple linear algebraic group over a field $k$. In Springer's Linear Algebraic group, the Fundamental Group of $G$ is defined by a certain quotient of groups $P/Q$, which ...
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Kernel of ${\rm GL}(n,F)$ on ${\rm PG}(n-1,F)$ over a division ring $F$

I am reading Peter Cameron's note on Classical Groups and I got confused with Proposition 2.1 on page 14. I have no problem in proving that the elements in kernel are scalars. However, I don't ...
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1 answer
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A description of the compact symplectic group

Let $\mathrm{Sp}(2m;\mathbb{C})=\{X\in\mathrm{GL}(2m;\mathbb{C});X^t\Omega X=\Omega\}$, where $\Omega=\begin{bmatrix}0& I_m\\ -I_m& 0\end{bmatrix}$, $I_m$ is $m\times m$ identity matrix. The ...
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1 answer
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Why must these have integer coefficients?

We are considering diagonal subgroup of classical groups and their lie algebras. We then consider $l=a_1l_1 + a_2l_2 + ...$ where $l_i(H)$, H in the lie algebra, returns the ith entry of H. We then ...
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Classical groups - applications?

Are there any applications of classical groups in subjects like algebraic geometry, algebraic number theory, algebraic topology or arithmetic geometry? If there is, then can anyone please give some ...
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1 answer
298 views

Integer Cech cohomology of $SU(3)$

I am interested in computing the Cech cohomology (with integer coefficients) of the group $SU(3)$. I particularly care about $H^k(SU(3),\mathbb{Z})$ with $k=7$, although ideally I would like to be ...
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5 votes
0 answers
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Centralizer of $U(n)$ inside $U(nm)$

Let $n$ and $m$ be two positive integers. There is a canonical inclusion $U(n) \rightarrow U(nm)$ given by the tensor product with the unit matrix $\mathbf{1}_m$. What is the centralizer of $U(n)$ ...
Antoine's user avatar
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1 vote
0 answers
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Centraliser of orthogonal group in general linear group

I'm looking for a reference that tells me either the centraliser of $\mathrm{O}^\epsilon(n,q)$ in $\mathrm{GL}(n,q)$, or how to describe the module homomorphisms of $\mathrm{O}^\epsilon(n,q)$ acting ...
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