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Component groups of stabilizers for linear representations

Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$. Given a vector $v \in V$, it is natural to consider its stabilizer group $...
Zhiyu's user avatar
  • 6,622
9 votes
2 answers
865 views

Multiplication in Peter-Weyl theorem

$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
Yellow Pig's user avatar
  • 2,964
1 vote
1 answer
114 views

A correspondence between projective representations of $G$ with those of its universal cover

Let $G$ be a connected Lie group and $\mathcal{H}$ be a Hilbert space. Let $U(\mathcal{H})$ denote the the group of all unitary operators on $\mathcal{H}$ with function composition (i.e., $\hat{U}:\...
Mahtab's user avatar
  • 287
0 votes
0 answers
68 views

A reference for this statement (representations of universal central extensions)

Let $G$ be a connected Lie group with universal cover $\tilde{G}$. I would really appreciate if someone could give me a reference for the proof of the following fact: "Every projective unitary ...
Mahtab's user avatar
  • 287
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
6 votes
1 answer
352 views

All surjections onto trivial irrep split equivalent to being reductive

$\DeclareMathOperator\Hom{Hom}$Let $ G $ be linear algebraic group over a field $ k $. Is it true that every short exact sequence of algebraic $ G $-representations $$ 0 \to W \to V \to k \to 0 $$ ...
Ian Gershon Teixeira's user avatar
8 votes
1 answer
534 views

Representation theory of $\mathrm{GL}_n(\mathbb{Z})$

I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
Kenji's user avatar
  • 81
6 votes
0 answers
149 views

Algorithmic representation of the Spin (and Pin) group [duplicate]

Performing algorithmic computations in $\mathit{SO}_n(\mathbb{R})$ or $\mathit{O}_n(\mathbb{R})$ is easy: its elements are represented by $n\times n$ orthogonal matrices of reals so, assuming we have ...
Gro-Tsen's user avatar
  • 32.5k
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
1 vote
1 answer
423 views

Quaternion representation and Haar measure of $SU(3)$ [closed]

Do we have easily and practically useful quaternion representation for $SU(3)$ group element and for Haar measure? Also, is $SU(2)$ really simplified in the quaternion base?
Sergii Voloshyn's user avatar
6 votes
0 answers
201 views

Why should Serre's conjecture on congruence subgroup property hold?

There seem to be several related properties of an algebraic group, exhibiting the dichotomy between rank 1 and rank $\ge2$. Whether a lattice in the group satisfies the congruence subgroup property, ...
GTA's user avatar
  • 1,024
9 votes
1 answer
277 views

Algorithmically handling the Spin groups in larg(ish) dimensions

Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$? By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute ...
Gro-Tsen's user avatar
  • 32.5k
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
13 votes
1 answer
398 views

Mathematical explanation of orbital shell sizes: why is it sufficient to consider single-electron wave functions?

Motivation The question "Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?" asks for an explanation of the sequence 2, 8, 8, ...
Dan Romik's user avatar
  • 2,549
52 votes
2 answers
5k views

Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?

$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
Eugene Stern's user avatar
6 votes
1 answer
255 views

Which Lie groups are a central extension of an algebraic group?

Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an ...
Luis's user avatar
  • 161
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
1 vote
1 answer
137 views

Does every locally compact, simply connected group admit enough finite dimensional representations?

Given a simply connected locally compact group $G$, is it true that $G$ admits enough finite dimensional representations (over any field and not necessarily continuous) to separate points in $G$, what ...
Rick Sternbach's user avatar
3 votes
0 answers
205 views

Status of RFD groups and $C^*$-algebras

Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
Rick Sternbach's user avatar
2 votes
1 answer
191 views

Normalizer of SU$(2)$ in SU$(6)$

Consider the $\mathfrak{su}(2)$ subalgebra of $\mathfrak{su}(6)$ embedded as $$\mathfrak{su(2)}=\text{Span}\{\mathbb{1}_3 \times \sigma^i\}, \quad i=1,2,3$$ with $\sigma^i$ the Pauli matrices and $\...
Rudyard's user avatar
  • 155
2 votes
1 answer
220 views

Set of $\mathrm{SU}(6)$ matrices which conjugate $\mathbb{1}_3 \otimes \sigma^3$ subalgebra element into $\mathfrak{su}(2)$

$\DeclareMathOperator\SU{SU}$Consider the Lie group $\SU(6)$, its Lie algebra $\mathfrak{su}(6)$ and the $\mathfrak{su}(2)$ subalgebra spanned by $\mathbb{1}_3 \otimes \sigma^i$, where $\sigma^i$ are ...
Rudyard's user avatar
  • 155
2 votes
1 answer
114 views

Set of $U(6)$ elements which leave a Lie-algebra element invariant under conjugation

Consider the specific element of the corresponding Lie algebra $\mathbb{1}_3 \times \sigma^3$, where $\mathbb{1}_3$ is the unit matrix in 3 dimensions, $\sigma^3$ is the 3rd Pauli matrix and $\times$ ...
Rudyard's user avatar
  • 155
9 votes
2 answers
523 views

Is the group of integer points of a simple real linear algebraic group a maximal closed subgroup?

Let $ G $ be a simple linear algebraic group. Let $ G_\mathbb{R} $ be the real points of $ G $. Let $ G_\mathbb{Z} $ be the integer points of $ G $. Is $ G_\mathbb{Z} $ a maximal closed subgroup? In ...
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
1 vote
1 answer
204 views

Injective group homomorphism on $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/2}$ or $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/4}\to U(2^{2k})$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$From Pierre Deligne's Notes on spinors, we can see that there ...
Марина Marina S's user avatar
12 votes
1 answer
655 views

Which Lie groups have finitely many conjugacy classes of subgroups of fixed isomorphism type?

Let $G$ be a real Lie group. What conditions must $G$ satisfy so that the following is true: For any finite group $\Gamma$ there exist finitely many conjugacy classes of subgroups of $G$ that are ...
user505117's user avatar
6 votes
2 answers
401 views

Relations between $3j$-symbols and intertwiners

I am trying to understand the relation between Wigner's $3j$-symbols (or Clebsch-Gordan coefficients) and matrix coefficients of intertwiners. I am new to this topic and need some help to understand ...
G. Blaickner's user avatar
  • 1,429
11 votes
1 answer
589 views

A detail in the proof of Schur's lemma: the closures of the $\mathcal{Ker}$ and $\mathcal{Im}$ of the intertwiner

$\renewcommand\Im{\operatorname{\mathcal{Im}}}\newcommand\Ker{\operatorname{\mathcal{Ker}}}$I was sure that this is a trivial question and placed it on Math Stackexchange https://math.stackexchange....
Michael_1812'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
3 votes
0 answers
547 views

Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$

Given a group $G$, we denote the center Z$(G)$, we like to know the automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences: $$...
wonderich's user avatar
  • 10.5k
8 votes
2 answers
482 views

Parabolics and simple roots for a special unitary group: reference request

I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed. ...
Mikhail Borovoi's user avatar
2 votes
0 answers
81 views

The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin} $Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$. Below I specify a specfic way to embed $...
wonderich's user avatar
  • 10.5k
2 votes
0 answers
111 views

The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $(\Spin(...
wonderich's user avatar
  • 10.5k
3 votes
1 answer
355 views

The normalizer of SU(n) in U(m)?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $...
wonderich's user avatar
  • 10.5k
1 vote
1 answer
275 views

The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?

$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that $$ \U(2^{N-1})\supset \Spin(2N) $$ when $2N > 4$ or a positive integer $N > 2$, so $\Spin(2N)$ can be embedded in $\U(2^...
wonderich's user avatar
  • 10.5k
5 votes
1 answer
420 views

Analogue of the special orthogonal group for singular quadratic forms

The special orthogonal group $SO(n)$ is the subgroup of the special linear group $SL(n)$ of $n\times n$ matrices with determinant one that preserve a non-degenerate symmetric bilinear form. If such a ...
user avatar
5 votes
0 answers
298 views

What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?

What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$? Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\...
Andrew NC's user avatar
  • 2,071
15 votes
2 answers
2k views

Why are coroots needed for the classification of reductive groups?

As we know reductive groups up to isomorphism corresponds to root data up to isomorphism. My question is why in the definition of root data do we need the coroots? Let's break it down to two questions:...
Andrew NC's user avatar
  • 2,071
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
4 votes
1 answer
321 views

Average of product of matrix elements in the special orthogonal group

Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed, in Integration with respect to the Haar measure on unitary, orthogonal and symplectic group (see ...
Marcel's user avatar
  • 2,552
1 vote
1 answer
133 views

Irreducible non-Abelian subgroup of $\mathrm{U}_n(\mathbb{C})$, containing diagonal matrices

Consider an irreducible non-Abelian subgroup $\mathrm{H}$ of group of unitary matrices $\mathrm{U}_n(\mathbb{C})$, that contains the subgroup of diagonal matrices. Does there exist any result ...
Mini's user avatar
  • 85
3 votes
1 answer
151 views

Morphisms of two fully reducible representations of a group [closed]

I thought this is a simple question and placed it at math.stackexchange.com: https://math.stackexchange.com/questions/3616661/morphisms-of-two-fully-reducible-representations-of-a-group. Since no one ...
Michael_1812's user avatar
4 votes
0 answers
68 views

The weak restriction of the Jacquet module

Let $P= MN$ be a parabolic subgroup of a reductive p-adic group $G$, and $(\pi, V)$ is an irreducible, admissible representation of $G$. The Jacquet module is the representation $(\pi_N, V_N)$, where $...
IMED's user avatar
  • 41
6 votes
2 answers
280 views

How to check whether a given matrix is in the image of a representation?

Let $G$ be a compact simple Lie group, and let $\rho$ be a (faithful, unitary) irreducible representation thereof of $\mathbb K$-dimension $n$, where $\mathbb K=\mathbb C/\mathbb R/\mathbb H$ if $R$ ...
AccidentalFourierTransform's user avatar
3 votes
0 answers
167 views

Representation R where the center of Spin group acts trivially on R

For the following groups, I would like to know the given this group G and its representation R such that the center of G acts trivially (i.e. acts nothing) on R. Let us denote $\operatorname{Spin}(n,\...
annie marie cœur's user avatar
4 votes
2 answers
428 views

A morphism intertwining two induced representations

TL;DR: Given representations $D,\Lambda$ of subgroups $K,Q$ of a Lie group $G$, is it true that every intertwining operator $T$ between the resulting induced representations of $G$ can be written $$ (...
Michael_1812's user avatar
23 votes
2 answers
611 views

Does the 3875-dimensional rep of $E_8$ have a solution to $x\star x=0$?

Consider the compact Lie group $E_8$. Its second-smallest fundamental representation is $3875$-dimensional and admits a symmetric invariant form, and so is real: $E_8 \curvearrowright \mathbb{R}^{3875}...
Theo Johnson-Freyd's user avatar
7 votes
1 answer
2k views

Automorphism group of the special unitary group $SU(N)$

Let us consider the automorphism group of the special unitary group $G=SU(N)$. We know there is an exact sequence: $$ 0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0. $$ For $G=SU(2)...
annie marie cœur's user avatar
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
7 votes
1 answer
237 views

Finite subgroups of $PSU(3)$

I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
Tarik's user avatar
  • 71