All Questions
Tagged with finite-groups lie-groups
61 questions
1
vote
1
answer
274
views
The compact Lie group contains a finite subgroup $\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$
Given a finite Abelian group: $G=\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$, where ${n_1},{n_2},{n_3}$ are arbitrary positive integers. ${n_1},{n_2},{n_3}$ may have or may not ...
- 10.5k
2
votes
0
answers
134
views
Finite subgroups of compact simple Lie groups [duplicate]
The finite subgroups of $SU(2)$ consist of the symmetry groups of the Platonic solids plus the finite subgroups of $O(2)$. I would like to know if there are any results concerning $SU(3)$. In ...
- 351
8
votes
2
answers
2k
views
The number of conjugacy classes of the simple group PSL(2,q)
If $q=p^a$ , where $p$ is a prime number, then I would like to know the number of conjugacy classes related to elements of order $p$ and $2$ in the simple group $PSL(2,q)$ .
- 169
9
votes
2
answers
634
views
Extension of the Weyl dimension formula
Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...
- 2,446
1
vote
1
answer
104
views
The action of graph automorphism of finite symplectic group on maximal subgroups
Let $G=Sp(4,2^f)$ with $f>1$. Based on the facts when $f$ is small, I would feel the following:
$G$ has two conjugacy classes of subgroups isomorphic to $SO^+(4,2^f)$. One is in Aschbacher's class ...
- 767
11
votes
2
answers
1k
views
Finite subgroups of $PGL(3,K)$
It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
- 125
2
votes
0
answers
123
views
Finite subgroups (lattices) in the large N limit of SU(N)
I would like to gain some information about the discrete subgroups (lattices) of SU(N) Lie groups. I have already read some answers and references concerning the N=3 and N=4 cases. I am more ...
- 21
3
votes
0
answers
572
views
How to find the normalizer of a finite subroup in a Lie group?
If a group $G$ is generated by finitely many subgroups $G_i$ and $H$ a subgroup of $G$, under which conditions can $N_G(K)$, the normalizer of $K$ in $G$, be generated by all the normailizers $N_{G_i}(...
- 31
11
votes
2
answers
491
views
Cohomology of $T^n/W$ for compact Lie groups
Let $G$ be a compact, connected and simply connected Lie group.
Let $T\subset G$ be a maximal torus and let $W$ be the corresponding
Weyl group.
Then we have the diagonal action of $W$ on $T^{n}$ ...
18
votes
3
answers
3k
views
Which groups have only real and quaternionic irreducible representations?
Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:
1) it's not isomorphic to its dual (...
- 22.3k
14
votes
3
answers
1k
views
Restriction from $GL_n$ to $S_n$
Let $V$ be the irreducible representation of $GL_n$ with highest $\lambda$, and $|\lambda|=n$. It is well known that the representation of $S_n$ on the $(1,1,\ldots,1)$ weight space is the Specht ...
- 156k