Recently Active 'finite-groups+lie-groups+rt.representation-theory' Questions

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

4,081

modified Dec 4 at 16:17

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, ...

Aurel

5,382

modified Oct 2 at 13:37

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=\...

LSpice

12.9k

modified Sep 6 at 0:08

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 $C_G(g)$ ...

Kenta Suzuki

3,054

modified Aug 13 at 12:10

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 ...

Maarten Havinga

1,245

modified May 11 at 11:44

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 ...

LSpice

12.9k

modified Feb 15 at 22:18

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 ...

Paul Levy

1,336

answered Nov 2, 2023 at 16:39

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

4,081

asked Aug 7, 2023 at 22:28

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 ...

YCor

63.9k

answered Jun 30, 2023 at 8:11

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 ...

Dave Benson

16.2k

modified May 1, 2023 at 18:58

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

4,081

modified Oct 12, 2022 at 22:43

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

4,081

modified Sep 26, 2022 at 12:27

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

4,081

modified Apr 7, 2022 at 2:37

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 $ ...

Venkataramana

11.2k

modified Mar 9, 2022 at 0:42

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 ...

LeechLattice

9,501

modified Apr 25, 2021 at 16:54

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 ...

LSpice

12.9k

modified Jul 28, 2020 at 18:52

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

41

asked Apr 22, 2019 at 13:49

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 ...

Richard Stanley

50.8k

modified Oct 31, 2018 at 8:40

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

163

modified Sep 21, 2018 at 3:33

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

61

modified Feb 16, 2018 at 17:54

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 $H$ ...

CommunityBot

1

modified Feb 9, 2018 at 9:12

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 ...

vytas

41

answered Jan 21, 2018 at 15:52

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 ...

Paul

501

modified Jun 11, 2015 at 0:41

1 vote

1 answer

167 views

Finite groups normalizing a torus

Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...

Will Sawin

148k

answered May 11, 2015 at 19:54

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 $\...

Nick Gill

11.2k

modified Aug 16, 2014 at 1:46

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, ...

CommunityBot

1

modified Apr 11, 2014 at 13:41

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 ...

Dan Petersen

40.2k

Aug 14, 2013 at 19:27

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 (...

CommunityBot

1

modified Nov 28, 2010 at 22:30

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All Questions

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

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

Normalizer of connected subgroup contained in the Weyl group?

Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$

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

Is the monster group maximal in SO(196883)?

Homomorphisms from binary polyhedral group to compact Lie groups

Are the integer points of a simple linear algebraic group 2-generated?

Compact Lie group has finitely many Lie primitive subgroups

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

Finite maximal closed subgroups of Lie groups

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

Aschbacher classes for compact simple group

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

Is the representation of finite simple groups fully understood?

Explicit tensor product decomposition for the representations of PSL(2,q)

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

Littlewood Richardson Rule for general linear group over finite field

Possible dimensions for triples of unitary irreducible representations whose tensor product contains the identity

Number of orthogonal operators in representations of the Unitary Group

Multiplicative subgroups of $GL(V)$ which are almost additively closed

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

Restriction from $GL_n$ to $S_n$

Finite groups normalizing a torus

Special linear groups contained in symplectic groups

Connection between two theorems on character values?

Extension of the Weyl dimension formula

Which groups have only real and quaternionic irreducible representations?

All Questions

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