Skip to main content

All Questions

Filter by
Sorted by
Tagged with
4 votes
1 answer
441 views

Large(ish) finite non-abelian subgroups of $\operatorname{GL}_n \mathbb C$ for $n>70$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SU{SU}\newcommand{\C}{\mathbb{C}}$My question is about large order finite non-abelian subgroups of $\GL_n\C$ without an ...
Fetchinson0234's user avatar
11 votes
1 answer
331 views

A question on groups having a subgroup which fixes a vector in every irreducible representations

Given a finite group $G$, I am interested in finding a non-trivial proper subgroup $H$ of $G$ such that $\mathrm{Ind}_H^G\mathbf{1}$ contains all the irreducible representations of $G$, that is, ...
Soumyadip Sarkar's user avatar
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=\...
Ian Gershon Teixeira's user avatar
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 $C_G(g)$ ...
kindasorta's user avatar
  • 2,907
3 votes
1 answer
268 views

Finite-maximal subgroups of orthogonal groups

I define a finite subgroup $H$ of a group $G$, finite-maximal if for any $g\in G\setminus H$, $\langle H,g\rangle$ is infinite. My question is now to find the finite-maximal subgroups of $\mathrm{SO}...
Andrea Aveni's user avatar
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 ...
Ian Gershon Teixeira's user avatar
5 votes
1 answer
653 views

What are the maximal closed subgroups of $ \operatorname{SU}_3 $?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$What are the maximal closed subgroups of $ \SU_3 $? This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
9 votes
2 answers
831 views

Signed permutations in $ \operatorname{SO}(n) $ and normalizing an extraspecial group

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Lift{Lift}$The subgroup of $ \SO(n) $ of signed permutations has order $ n!2^{n-1} $. I will call ...
Ian Gershon Teixeira's user avatar
10 votes
2 answers
914 views

Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$

Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers? I can only find ...
Mare's user avatar
  • 26.5k
0 votes
1 answer
183 views

Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\...
Andrea Aveni's user avatar
6 votes
1 answer
368 views

Number of points on a linear algebraic group over a finite field

Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be a good, simple lower bound for $G(F_q)$? One can get something fairly nice ...
H A Helfgott's user avatar
  • 20.2k
6 votes
1 answer
303 views

Compact Lie group has finitely many Lie primitive subgroups

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$A closed subgroup $ \Gamma $ of a Lie group $ G $ is called Lie primitive if it is not contained in any proper ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
95 views

Image of minimal degree representation of quasisimple group unique up to conjugacy

Let $ G $ be a quasisimple finite group. Let $ d_{min} $ be the minimum dimension of a nontrivial irrep of $ G $. Must it be the case that the image of all (nontrivial) dimension $ d_{min} $ irreps of ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
11 votes
1 answer
357 views

Alternating subgroups of $\mathrm{SU}_n $

$\DeclareMathOperator\PSU{PSU}$Let $ \PSU_n $ be the projective unitary group. Let $ A_m $ be the alternating group on $ m $ letters. $ A_5 $ is a maximal closed subgroup of $ PSU_2 \cong SO_3(\mathbb{...
Ian Gershon Teixeira's user avatar
6 votes
1 answer
567 views

Finite simple groups and $ \operatorname{SU}_n $

A follow-up question to Alternating subgroups of $\mathrm{SU}_n $. $\DeclareMathOperator\PU{PU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PU_m $ be the projective unitary group, ...
Ian Gershon Teixeira's user avatar
7 votes
2 answers
1k views

Finite subgroups of $\operatorname{U}(2)$

Famously, the finite subgroups of $\operatorname{SU}(2)$ admit an ADE classification. Question. Is there a similar result for finite subgroups of $\operatorname{U}(2)$? Are they classified? If this ...
Federico Carta's user avatar
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 ...
emiliocba's user avatar
  • 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 $\mathrm{SL}_2(\mathbb{H})$?
Daniel Sebald's user avatar
6 votes
1 answer
445 views

Is every finite subgroup the integer points of a linear algebraic group?

Cross Posting this from MSE since it's been there for almost a month and it got a couple upvotes but no answers. MSE link Is every finite subgroup the integer points of a linear algebraic group? Let $ ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
383 views

$ S_4 $ subgroups and $ \operatorname{SO}_3(\mathbb{R}) $

$\DeclareMathOperator\SO{SO}$I posted this on MSE 10 days ago and it got 3 upvotes but no answers or comments, so I'm cross-posting to MO. Background: The group of rotations $ \SO_3(\mathbb{R}) $ has ...
Ian Gershon Teixeira's user avatar
5 votes
1 answer
508 views

Finite maximal closed subgroups of Lie groups

Cross-posted from MSE https://math.stackexchange.com/questions/4272017/finite-maximal-closed-subgroups-of-lie-groups $\newcommand{\G}{\mathcal{G}} \newcommand{\K}{\mathcal{K}} \DeclareMathOperator\SU{...
Ian Gershon Teixeira's user avatar
4 votes
1 answer
237 views

Aschbacher classes for compact simple group

Posted this to MSE several weeks ago and it got 3 upvotes but no answers or even comments so I'm cross-posting to MO Aschbacher's theorem says that every maximal subgroup of a finite simple classical ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
365 views

Self-normalizing implies maximal for subgroup of compact Lie group

Consider the compact group $ G=\operatorname{SO}_3(\mathbb{R}) $. The closed subgroups of $ G $ (other than the trivial group 1 and the whole group $ G $) are exactly $ O_2$, $\operatorname{SO}_2 $ ...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
503 views

Is the representation of finite simple groups fully understood?

Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...
Dick Johnson's user avatar
0 votes
1 answer
130 views

Number of reduced decompositions of the dihedral group $D_6$ [closed]

The Weyl group of $\frak{g}_2$ is the dihedral $D_6$. Let us denote its longest element by $w_0$. How many reduced decompositions does $w_0$ have?
Martim Pereir's user avatar
4 votes
1 answer
633 views

Homomorphisms from binary polyhedral group to compact Lie groups

Are homomorphisms from binary polyhedral groups to (simple and simply connected) compact Lie groups classified? For cyclic groups, the result is well known (see e.g. Kac's "Infinite dimensional Lie ...
Yuji Tachikawa's user avatar
8 votes
1 answer
208 views

Can the defining rep of $E_7$ split over a finite subgroup while the adjoint remains simple?

Does the (simply connected compact) Lie group $E_7$ contain a finite subgroup $G \subset E_7$ such that the $56$-dimensional irrep of $E_7$ splits over $G$ as $28 \oplus \overline{28}$, but the $133$-...
Theo Johnson-Freyd's user avatar
8 votes
2 answers
618 views

Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field

Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality. Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$? I understand ...
Q-Zh's user avatar
  • 960
10 votes
5 answers
3k views

Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...
user47245's user avatar
  • 101
2 votes
0 answers
164 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 ...
Sebastien Palcoux's user avatar
15 votes
1 answer
784 views

The completion of the space of finite groups

Edit: I revise the question based on the comment conversations Let $\mathcal{F}$ be the set of all equivalence classes of finite groups under the "Isomorphism" equivalence relation. We define ...
Ali Taghavi's user avatar
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 ...
Simon Lentner's user avatar
3 votes
1 answer
298 views

Littlewood Richardson Rule for general linear group over finite field

I just finished reading Green's 1955 paper on characters of general linear groups and have also been reading Macdonald's Symmetric Functions and Hall Polynomials. I see that there is a recursive ...
Sheel Stueber's user avatar
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 (...
John Baez's user avatar
  • 22.3k
2 votes
0 answers
200 views

What are the points about representation of groups? [closed]

For a fixed (let say finite-, or Lie-, to respect the historical motivations) group, why does the study of all its linear representations over a fixed field, leads to some knowledge about its ...
huurd's user avatar
  • 41
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 $...
Shawn's user avatar
  • 453
23 votes
1 answer
1k views

Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups. Below is a proposition for ...
Joonas Ilmavirta's user avatar
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 ...
David E Speyer's user avatar
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 $PSL(2,q)$ .
Tina's user avatar
  • 169
4 votes
1 answer
189 views

cohomology of finite groups of lie type with coefficients in the adjoint module

Let $\mathbb G$ be a connected, semisimple, split group over a finite field $\mathbb F_q$ and let $G = \mathbb G(\mathbb F_q)$. Let $\mathfrak g$ be its Lie algebra, an $\mathbb F_q$-vector space with ...
user94041's user avatar
  • 391
5 votes
1 answer
2k views

Weyl groups of $E_6$ and $E_7$

The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...
Hebe's user avatar
  • 951
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 ...
Mark's user avatar
  • 163
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 $T^{n}$ ...
José Manuel Gómez's user avatar
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 ...
Tim's user avatar
  • 125
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 $H$ ...
Ali Taghavi's user avatar
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 ...
emiliocba's user avatar
  • 2,446
5 votes
1 answer
549 views

Special linear groups contained in symplectic groups

Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and $\...
Huangjun Zhu's user avatar
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 ...
benkj's user avatar
  • 61