All Questions
13 questions
9
votes
0
answers
210
views
Why and how is a representation "continuously decomposable"?
What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...
- 5,424
2
votes
2
answers
230
views
Does the map $f \mapsto \mu_f$ (BV to Lebesgue-Stieltjes measure) behave nicely under function concatenation?
Consider two continuous functions $f,g : [0,1]\rightarrow\mathbb{R}$ of bounded variation, and let $\mu_f, \mu_g : \mathcal{B}([0,1])\rightarrow\mathbb{R}$ be their associated Lebesgue-Stieltjes (...
- 463
10
votes
2
answers
594
views
Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
- 960
4
votes
1
answer
155
views
Resource on spectral theory for differential operators with symmetry groups
In Methods of Mathematical Physics IV by Reed and Simon, the authors cover Floquet theory in detail in Section XIII.16. On page 280, they note that
"A part of the analysis of [the periodic ...
- 854
1
vote
0
answers
50
views
Representability of smooth invertible Lipschitz functions by a finite composition of near-identity functions
Theorem 1 of this paper shows that
For every positve integer $K$ and for every nonempty bounded domain $\mathcal X \subseteq \mathbb R^d$, the restriction on $\mathcal X$ of any function $h: \...
- 6,853
4
votes
1
answer
615
views
Representation of Heisenberg-Weyl elements and their exponentials
There is possibly a huge literature on the subject but I am a newcomer on analytic representations and my need is rather specific. I simplify it below.
Let $A,B$ be two symbols (standing for ...
- 3,738
2
votes
0
answers
72
views
"Spectral gaps" of commutativity measure
There's a notion of commutativity measure $P(G)$ of a finite group $G$ which is probably folklore: count commuting pairs in $G \times G$ and divide by $|G \times G|$. There are some results:
$P(...
- 4,600
5
votes
0
answers
215
views
Explicit description of the Plancherel measure for $GL_n(\mathbb{R})$
Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation
$$f(g)...
- 1,692
10
votes
1
answer
802
views
Restriction of irreducible unitary representation to normal subgroup of finite index
Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...
- 333
6
votes
1
answer
227
views
Quantum group representations from (convolution) matrix units?
Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...
- 1,037
4
votes
1
answer
267
views
reference request: direct product of WOT-continuous unitary representations
In an article I'm revising, I spend some time giving a self-contained proof of the following result
Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary ...
- 25.8k
19
votes
3
answers
1k
views
Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
- 965
8
votes
1
answer
920
views
Looking for references talking about category of topological vector spaces
It's known that category of topological vector spaces is not abelian but quasi-abelian or exact category. I am looking for the references playing with this category(category theory). All the related ...
- 5,515