All Questions
96 questions
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 ...
- 541
27
votes
3
answers
3k
views
Is there a 'nice' interpretation of virtual representations?
Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...
- 5,191
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}...
- 54.6k
22
votes
1
answer
720
views
Does $E_8$ know $Spin(7)$?
One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For example,...
- 54.6k
17
votes
2
answers
597
views
When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$?
A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, ...
- 2,851
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:...
- 2,071
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 ...
- 657
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
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, ...
- 2,549
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 ...
- 577
11
votes
2
answers
1k
views
Realizing a subgroup of a Lie group as a stabilizer subgroup
Let $G$ a compact semisimple Lie group, $H$ a subgroup of $G$. Is it always possible to find an irreducible representation $R$ of $G$ such that the stabilizer of an $x\in R$ is "locally isomorphic" to ...
- 679
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....
- 603
9
votes
2
answers
834
views
Lattices in $SL(n,\mathbb R)$
If $\Gamma\subseteq SL(n,\mathbb{R})$ is a lattice (i.e. discrete and finite covolume), does $\Gamma$ necessarily contain some $\mathbb{R}$-diagonalizable copy of $\mathbb{Z}^{n-1}$?
I know that the ...
- 93
9
votes
2
answers
522
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 ...
- 4,081
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 ...
- 2,964
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 ...
- 32.5k
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 ...
- 81
8
votes
3
answers
2k
views
Splitting of a Short-Exact Sequence of Lie Groups
Let $G$ be a Lie group (possibly disconnected). Consider the natural short-exact sequence $$1\rightarrow G_0\rightarrow G\rightarrow\pi_0(G)\rightarrow 1,$$ where $G_0$ is the identity component of $G$...
- 4,920
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.
...
- 14.2k
8
votes
1
answer
929
views
Raising and lowering operators for SL(n,K) on homogenous polynomials
The short version:
Can the theory of weights for SL(n,C) be explained concretely in terms of raising and lowering operators on spaces of polynomials?
A deleted question asked how to prove SL(3,C) ...
- 10.7k
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)...
- 2,147
7
votes
3
answers
2k
views
Characterising the adjoint representation of SU(N)
One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an $...
- 71
7
votes
1
answer
456
views
Can Galois conjugates of lattices in SL(2,R) be discrete?
Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
- 454
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?
- 71
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
6
votes
2
answers
400
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 ...
- 1,429
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 ...
- 977
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
1
answer
1k
views
Decomposition of semisimple Lie group into almost simple factors
Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk ...
- 511
6
votes
1
answer
390
views
Can these two irreducible $GL_n \mathbb Z$-representations be isomorphic?
Fix $n\in \mathbb N$ and a partition $\lambda$ with at most $n-1$ parts (of length at most $n-1$). Let $V$ be the irreducible $GL_n \mathbb R$-representation with highest weight $\lambda$ and $D$ the ...
- 236
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 $R$ ...
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 ...
- 161
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 ...
- 32.5k
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
5
votes
2
answers
452
views
"geometric" description of the algebra of central functions on a Lie group
I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
- 2,201
5
votes
1
answer
647
views
Orbit structure of linear representations of complex Lie groups
Let $G$ be a semisimple complex Lie group (or perhaps a reductive algebraic group over $\mathbb{C}$) and $V$ an irreducible finite-dimensional representation of $G$, determined by its highest weight. ...
- 51
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 ...
- 17.9k
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 ...
user82886
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 $\...
- 533
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
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_{\...
- 2,071
5
votes
0
answers
214
views
What do we know about isospectral Cayley graphs?
Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...
- 1,893
4
votes
2
answers
578
views
Proper compact connected subgroup of $Spin(n)$
What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am ...
- 471
4
votes
1
answer
1k
views
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 ...
- 1,719
4
votes
2
answers
363
views
Complexification or 'real'ization of Mapping Class group.
So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
- 327
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
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
$$
(...
- 603
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 ...
- 2,552
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 ...
- 6,123
4
votes
1
answer
1k
views
Reductive Lie Groups and Complexification
Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, $G$...
- 4,920