All Questions
96 questions
1
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71
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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 $...
- 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 ...
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 ...
- 2,368
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}:\...
- 3,054
0
votes
0
answers
68
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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 ...
- 287
4
votes
1
answer
1k
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How to think about the simple reflection $s_0$ in the affine Weyl group?
Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
- 3,065
14
votes
0
answers
527
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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 ...
- 12.9k
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
$$
0 \to W \to V \to k \to 0
$$
...
- 4,081
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}...
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 ...
- 63.9k
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 ...
- 1,336
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 ...
- 1,574
6
votes
0
answers
149
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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 ...
- 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 ...
- 4,081
2
votes
3
answers
318
views
Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra
Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...
- 44.7k
1
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1
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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?
- 188k
15
votes
1
answer
656
views
Linear embeddings of nilpotent pro-$p$ groups
Is it true that every finitely generated (topologically) torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
- 63.9k
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{...
- 4,081
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, ...
- 4,081
6
votes
0
answers
200
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,
...
- 1,024
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 ...
- 4,081
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, ...
- 188k
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 ...
- 4,217
6
votes
1
answer
925
views
Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Let me firstly clarify my question. For example, we ...
- 1
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 ...
- 12.9k
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 ...
- 1,586
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 ...
- 15.5k
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 $\...
- 11.2k
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 $\times$ ...
- 31.5k
5
votes
2
answers
390
views
Dimensions of Jordan blocks associated to representations
Given a linear representation $\rho$ of $SL_n(\mathbb C)$ of finite dimension $m$,
the image $\rho(U)$ of a maximal unipotent Jordan block $U\in SL_n$ decomposes
into generally several Jordan blocks ...
- 6,349
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 ...
- 11.2k
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 ...
- 317
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 ...
- 11.6k
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 ...
- 63.9k
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....
- 20.9k
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 ...
- 9,501
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:
$$...
- 10.5k
2
votes
0
answers
81
views
The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}
$Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$.
Below I specify a specfic way to embed $...
- 12.9k
8
votes
2
answers
482
views
Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...
- 23k
3
votes
1
answer
355
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The normalizer of SU(n) in U(m)?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $...
- 148k
2
votes
0
answers
111
views
The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $(\Spin(...
- 10.5k
1
vote
1
answer
275
views
The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?
$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that
$$
\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^...
- 108k
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 ...
- 63.9k
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_{\...
- 63.9k
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:...
- 156k
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 ...
- 12.9k
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 ...
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 ...
- 63.9k
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 ...
- 12.3k
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 $...
- 41