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.
Questions tagged [classical-groups]
81 questions
20
votes
1 answer
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 statement $SL_n(K) \simeq SL_n(L) \implies K…
Ali Nesin
- 607
11
votes
2 answers
Intuitively, why are there 4 classical Lie groups/algebras?
I would like to understand the big picture in mathematics. Lie groups and Lie algebras seem to play a central role in bridging analysis and algebra. I'm curious to understand, intuitively, why there are 4 classical Lie groups/algebras…
Andrius Kulikauskas
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6
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2 answers
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" (in the sense of Lie algebra elements) in the simple…
theHigherGeometer
- 2,610
6
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1 answer
Which finite simple groups contain $PSL(2,q)$ for some $q\geq 4$?
Which nonabelian finite simple groups contain $PSL(2,q)$ for some $q$?
Obviously $PSL(2,q)$ themselves do. Also, as $PSL(2,4)\cong PSL(2,5)\cong A_5\subset A_n,\; n\geq 5$, alternating (nonabelian simple) groups do as well. I believe I have read…
Ben Blum-Smith
- 21,560
5
votes
2 answers
Determining if these surjections have sections
Let $\pi:\ \operatorname{GL}(2,k)\ \longrightarrow\ \operatorname{PGL}(2,k)$ be the canonical homomorphism, and pick some finite subgroup $G\subset \operatorname{PGL}(2,k)$. Then we have an exact sequence
$$1\ \longrightarrow\ \{\alpha I\mid \alpha…
Ben Blum-Smith
- 21,560
5
votes
0 answers
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 $U(n)$ seen as a subgroup of $U(nm)$?
Edit: I was asked…
Antoine
- 1,747
4
votes
1 answer
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 order of these groups. Order of Symplectic Group(4,2)…
4
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1 answer
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 $N_{G}(A)$ is $A.N$ where $N$ is a maximal subgroup of…
scsnm
- 1,321
4
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1 answer
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 $df:\mathbb{R}\to \mathfrak{sl}_2$. Since we have a basis $E,F,H$ (denoted $e,f,h$ in…
user676464327
- 428
- 4
- 14
4
votes
1 answer
$K/k$ 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 difficulties with some notations and notions used by Weil,…
P-addict
- 720
4
votes
0 answers
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 which are invariant under a $c$-Hermitian form on…
Riju
- 4,215
4
votes
1 answer
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 $Z = \{M | M=\alpha I, \alpha \in \Bbb{R}\}$.
Define a homomorphism $ \phi :…
roymend
- 535
4
votes
1 answer
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 $\mathrm{PSL}(2,7)$ has non-conjugate Hall subgroups of…
Groups
- 2,906
4
votes
1 answer
Spin(4,1) = Sp(1,1) isomorphism
I am interested in the exceptional isomorphism Spin(4,1) = Sp(1,1).
The correspondance is already mentioned here: spin group Spin(4,1) but the explicit isomorphism is not given.
I would like to know where this isomorphism comes from.
Vivien
- 43
4
votes
1 answer
Regarding the representation theory of $SL_2(\mathbf{R})$.
Dear friends of mathematics,
I have the following question for you.
(a) According to Wikipedia there is a unique irreducible (real??) $2$-dimensional representation of $SL_2(\mathbf{R})$, which must be the standard representation $i$ (for it is…
