All Questions
Tagged with finite-groups lie-groups
20 questions with no upvoted or accepted answers
14
votes
0
answers
527
views
Is the monster group maximal in SO(196883)?
$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
- 4,081
8
votes
0
answers
666
views
Approximating Lie groups by finite groups
How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$...
- 8,086
8
votes
0
answers
408
views
Connection between two theorems on character values?
In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
- 52.9k
5
votes
0
answers
217
views
Almost conjugate subgroups of compact simple Lie groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition:
Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...
- 2,446
4
votes
0
answers
134
views
Finite subgroups of $\mathrm{SL}_2(\mathbb{O})$
What are the finite subgroups (subloops that are groups) of $\mathrm{SL}_2(\mathbb{O})$? In particular, are there any not contained in a $\mathrm{SL}_2(\mathbb{H})$?
- 2,773
4
votes
0
answers
449
views
Normalizer of a split torus
Let $G$ be a connected reductive group split over a field $k$. Let $T$ be a maximal split torus of $G$. Consider $N_G(T)$, the normalizer of $T$ in $G$, we have $N_G(T)/T \cong W$, the Weyl group of $...
- 453
4
votes
0
answers
226
views
Possible dimensions for triples of unitary irreducible representations whose tensor product contains the identity
For which triples $\{A,B,C\}$ of positive integers does there exist a (finite or compact) group $G$ with unitary irreducible representations of dimensions $A$,$B$, and $ C$ whose tensor product ...
- 163
3
votes
0
answers
71
views
Conjugacy classes in reductive group under adjoint action of parabolic subgroup
Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results ...
- 1,846
3
votes
0
answers
128
views
Extension of Tits' theorem on groups with a BN-pair of rank ≥ 3
Tits has proved that a finite simple group $G$ with a BN-pair of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$ the Coxeter system. The subset lattice of the set $S$ is ...
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
2
votes
0
answers
97
views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 2,907
2
votes
0
answers
65
views
Are the integer points of a simple linear algebraic group 2-generated?
Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
- 4,081
2
votes
0
answers
163
views
Explicit tensor product decomposition for the representations of PSL(2,q)
$\DeclareMathOperator\PSL{PSL}$Let the type of the character theory of a finite group $G$ be the list $[[d_1,n_1], \dotsc, [d_k,n_k]]$ with $1=d_1 < \dotsb < d_k$ and $n_i$ the number of ...
2
votes
0
answers
55
views
Number of orthogonal operators in representations of the Unitary Group
Let $G={\rm SU}(d)$ be the unitary group and $\rho(g)$ an irreducible representation of $g\in G$ in a $D$ dimensional Hilbert space $V$. Let $e_i\in V$ be the diagonal matrix whose only non-zero ...
- 61
2
votes
0
answers
127
views
Multiplicative subgroups of $GL(V)$ which are almost additively closed
Edit:
According to comments of YCor and Vincent, I revise the question.I appreciate their comments:
Let $G$ be a group.
We say that $G$ is a special group if it is isomorphic to a subgroup $H$ ...
- 356
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
1
vote
0
answers
72
views
Normalizer of connected subgroup contained in the Weyl group?
Let $ G $ be a simple Lie group. Let $ H $ be a connected subgroup of $ G $ such that $ N(H)/H $ is finite. In such a case, is $ N(H)/H $ always a subgroup of the Weyl group of $ G $?
For $ G=\...
- 4,081
1
vote
0
answers
161
views
N(H)/H and the Weyl group
Let $ H $ be a connected subgroup of $ G=\mathrm{SU}(n) $ such that $ N_G(H)/H $ is finite. Is $ N_G(H)/H $ always a subgroup of the symmetric group $ \mathrm{S}_n $?
I just noticed this from the ...
- 4,081
1
vote
0
answers
181
views
Discrete group action on the sphere
Let $f$ be a continuous function on $S^3$ and let $\xi^{\perp}=\{x\in S^3:\,x\cdot\xi=0\}$
be a two-dimensional equator of $S^3$ orthogonal to the direction $\xi\in S^3$ (here $x\cdot\xi$ stands for a ...
0
votes
0
answers
110
views
Brauer characters of finite simple group $E_8(5)$
I would like to find the irreducible characters of the group $E_8(5)$ (mod 2)?
Can anyone help? (I am elementary in working with Brauer characters)
Many thanks
- 169