All Questions
11 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
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
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
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
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
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
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
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 ...
- 163
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 ...
- 203
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 ...
- 155
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 $\...
- 155