All Questions
Tagged with unitary-representations gr.group-theory
38 questions
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
2
votes
0
answers
118
views
What are the finite-dimensional irreducible unitary representations of $E(3)$?
Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by
$$E(3)=SO(3)\ltimes T(3)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
- 31
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}:\...
- 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 ...
- 287
3
votes
1
answer
251
views
Irreducible unitary representation of PSL(2,Z)
Do we already know the classification of the finite-dimensional irreducible unitary representations of the modular group $PSL(2,\mathbb{Z})=\mathbb{Z}/2*\mathbb{Z}/3$?
I'm particularly interested in ...
- 663
2
votes
0
answers
155
views
Finite dimensional unitary representations of the discrete Heisenberg group
Let $H(\mathbb{Z})$ be the discrete Heisenberg group. What are the finite dimensional irreducible unitary representations of $H(\mathbb{Z})$? Do they all arise from the coordinate-wise quotient map to ...
4
votes
1
answer
237
views
Existence of 'maximal' finite permutation groups?
Let $S(n)$ be the (unitary) matrix group of $n\times n$ permutation matrices. This is clearly a finite group of order $n!$. It is well known that we can add diagonal unitary matrices with any finite ...
- 155
1
vote
0
answers
106
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Are generalized symmetric groups maximal finite groups (in a certain sense)? - Part II, Loose Ends
Let $S(m,n)$ be the generalized symmetric group which is a wreath product of the cyclic group of order $m$, denoted here by $\mathbb{Z}_m$, and the symmetric group $S_n$. A standard unitary ...
- 155
7
votes
1
answer
556
views
Are generalized symmetric groups maximal finite groups (in a certain sense)?
Let $S(m,n)$ be the generalized symmetric group which is a wreath product of the cyclic group of order $m$, denoted here by $\mathbb{Z}_m$, and the symmetric group $S_n$. A standard unitary ...
- 155
9
votes
1
answer
434
views
Questions on the group $\mathrm{GL}(H)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}$Let $H$ be an infinite dimensional complex Hilbert space. Consider the group $\GL(H)$ of bounded invertible operators on $H$.
Question 1. I've ...
- 1,586
7
votes
1
answer
403
views
Induction and restriction of unitary representations
$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\Res{Res}$Given a locally compact group $G$ and a closed subgroup $H\subset G$,
let $\Rep(G)$ and $\Rep(H)$ denote their ...
- 43.2k
9
votes
1
answer
238
views
Kazhdan's property (T) for $\tilde{C}_2$-lattices
It is known that higher rank lattices have property (T) and also that lattices on 2-dimensional Euclidean buildings have property (T) provided the thickness $q+1$ of the building is large enough (...
- 2,050
5
votes
0
answers
292
views
Which tensor power of a given representation contains the trivial one?
If $R$ is an irreducible representation of a simple Lie-groups $G$ I assume there is always a lowest integer $n$ such that the tensor product representation $R \otimes R \otimes \ldots \otimes R$ (n ...
- 1,150
2
votes
0
answers
163
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 ...
4
votes
1
answer
313
views
Properties of the spectrum of the Koopman representation
Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$.
A function $\lambda\colon G\to \mathbb C$ is an ...
- 111
4
votes
0
answers
124
views
Systems of imprimitivity for irreducible subgroup of GU(n,q)
My question is similar to this one but about finite field case.
So, the set up is the following:
Let $G$ be $GU_n(q)$ acting on unitary space $(V, {\bf f})$, where $V=\mathbb{F}_{q^2}^n$ and ${\bf f}...
- 178
4
votes
1
answer
261
views
Uniform Roe algebra of virtually abelian group is type I C*-algebra?
Let $G$ be an arbitrary (discrete) group. It acts by left translation on $\ell^\infty(G)$. The uniform Roe algebra of $G$ is defined as the crossed product $\ell^\infty (G) \rtimes_{\mathrm{red}}G$.
...
- 741
3
votes
0
answers
99
views
Unitary matrices $p$-root of identity such that the Fourier transform matrices are $p$-root of identity
Take a prime number $p$ and $\omega=e^{2i\pi/p}$. Assume we have p complex matrices (in finite dimension $n$) $A_0, \dotsc, A_{p-1}$ such that $\forall i, A_i^p=I$.
Define the $p$ fourrier transform ...
- 583
10
votes
1
answer
974
views
Finite-dimensional faithful unitary representations of SL(2,Z)
Does $SL(2,\mathbb{Z})$ have a finite-dimensional faithful unitary representation? No such representation exists for $SL(2,\mathbb{R})$, but I don't see a reason why one shouldn't exist for $SL(2,\...
- 541
5
votes
1
answer
490
views
Systems of imprimitivity for unitary representations - reference request
Let $G$ be a finite subgroup of the group $U_d(\mathbb{C})$ of unitary transformations of $\mathbb{C}^d$. Suppose that $G$ acts irreducibly but is imprimitive, meaning that there is a nontrivial ...
- 4,776
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
3
votes
0
answers
237
views
Orthogonality relations for unitary representations of infinite (finitely generated) groups
Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...
- 333
11
votes
2
answers
779
views
Characters of permutation groups
Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of
permutations on an $N$-element set that have exactly $m$ cycles (counting
$1$-cycles). Then it is in the literature that the ...
- 4,679
11
votes
2
answers
537
views
Groups without property (T) but all finite quotients are expanders
What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G \...
- 4,432
1
vote
0
answers
139
views
Representation of finite group
Let $\Gamma$ be a finite index subgroup of $SL_2(\mathbb{Z})$. Let $\pi$ be a natural covering from $\Gamma\backslash\mathbb{H}$ to $SL_2(\mathbb{Z})\backslash\mathbb{H}$. Denote by $\text{Deck}(\pi)$ ...
- 11
2
votes
1
answer
1k
views
Decomposing a reducible representation of the unitary group
Consider the representation $L_U$ of the unitary group $U(n)$ on $L(\mathbb{C}^n)$ where $L_U$: $L(\mathbb{C}^n) \rightarrow L(\mathbb{C}^n)$ is a linear operator that $L_U M=U M U^{\dagger} $, $\...
- 23
5
votes
0
answers
321
views
Unitary representations of Tarski Monsters and other beasts
Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...
- 647
7
votes
2
answers
244
views
Can monomial representations induced from nonmonomial representations?
Let $H$ be a subgroup of $G$. Let $\rho$ be an irr representation of $G$ induced from an irr representation $\theta$ of $H$. It is well known that $\rho$ is monomial if $\theta$ is monomial. Is it ...
- 533
18
votes
2
answers
924
views
Which groups are the unitary group of a $C^*$-algebra
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
- 221
6
votes
1
answer
299
views
Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?
Let $G$ be a locally compact (second countable) group and let
$$
G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}.
$$
This is the kernel of the ...
- 687
7
votes
1
answer
737
views
Unbounded representations of groups
Let $H$ be a Hilbert space and $G$ be a finitely generated group. Let $\pi:G\rightarrow GL(H)$ be a representation.
A map $c:G\rightarrow H$ is called cocycle if $c(gh)=π(g)c(h)+c(g)$ for all $g,h$ ...
- 4,705
21
votes
3
answers
1k
views
Is a reductive adelic group a Type I group?
I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is ...
- 955
18
votes
2
answers
1k
views
Regarding Cayley Graphs of Property (T) Groups
A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of ...
- 7,057
10
votes
2
answers
941
views
Induced representations of topological groups
Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups)
If $G$ is a group ...
- 18.7k
10
votes
1
answer
2k
views
Unitary representations of the ax+b group: an accessible presentation
The "ax+b group" is the group of affine transformations of $\mathbb R$. It is a locally compact non unimodular group.
Its space of irreducible, continuous unitary representations has been described ...
- 9,486
5
votes
4
answers
2k
views
Finite-dimensional faithful representations of compact groups
Is it true that a compact group always has a faithful, finite-dimensional unitary representation? If not, are there any reasonably simple counter-examples?
I've done some research and know that every ...
- 95
2
votes
1
answer
174
views
Is there an abstract characterization of freeness in terms of additive unitary cocycles?
This question is very closely related to my other question here.
Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map $b:\...
- 7,057
13
votes
3
answers
694
views
Is there a characterization of free groups in terms of the unitary dual?
If $G$ is a countable discrete group, I'm curious if it is possible to decide whether $G$ is a free group only by looking at properties of $Rep(G)$, the collection of (equivalence classes of) strongly ...
- 7,057