# Questions tagged [unitary-representations]

The unitary-representations tag has no usage guidance.

173
questions

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### Density of irreducible matrix coefficients of a locally compact group

Let $G$ be a locally compact group and $I$ the set of matrix coefficient of irreducible unitary matrix coefficients of $G$. By Gelfand-Raikov's theorem and Stone-Weirestrass's theorem, for a compact $...

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### 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 ...

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### Complex representations of groups of invertible elements in finite local rings

Let $R$ be a finite local $\mathbb{F}_p$-algebra, and let $J$ be its Jacobson radical. Assume that $R/J\cong \mathbb{F}_p$, and assume that the socle of $R$ as an $R$-bimodule is one dimensional over $...

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### Is there a non-constant function on the sphere that diagonalizes all rotations simultaneously?

INTRODUCTION. I am teaching a course in Harmonic Analysis. In class, very often I find myself stressing out the fundamental property that the functions
$$
e_n(x)=\exp(2\pi i n x), \quad \text{where }\...

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### Unitary dual of universal cover

The universal covering group $G$ of $\mathrm{SL}_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\...

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### Subrepresentations and the induced map on Lie algebra cohomology

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$Setup: Let $G$ be the group $\GL(4, \mathbb{R})$, $B$ denotes the Borel subgroup consisting of upper triangular matrices and $P_{(2,2)}$ be the ...

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### Algorithm for finding the symmetries of a linear operator

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Let $V, W$ be finite dimensional complex vector spaces and $M\in \Hom(V, W)$ a full rank linear map. I want to see if there exists a Lie group ...

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### 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 ...

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### Faithfull compact representation of the discrete Heisenberg group

Let $H$ be the discrete Heisenberg group. Is there an injective homomorphism $\varphi \colon H \to U(n)$ for some $n$? What about injective homomorphism $\varphi \colon H \to O(n)$ for some $n$?

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### Is the left-regular representation of a locally compact group a homeomorphism onto its image?

Consider the left-regular representation $\lambda : G \to B(L^2(G))$, $\lambda_g f(h) = f(g^{-1}h)$, for a locally compact group.
It is well-known that this is a unitary faithful and strongly-...

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### Real Representation ring of $U(n)$ and the adjoint representation

I have two questions:
It is well known that the complex representation ring $R(U(n))=\mathbb{Z}[\lambda_1,\cdots,\lambda_n,\lambda_n^{-1}]$, where $\lambda_1$ is the natural representation of $U(n)$ ...

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### Classifying endomorphisms of a direct sum Hilberts pace

Suppose I have a Hilbert space with a direct sum structure into "superselection sectors", i.e. $\mathcal{H} = \oplus_\alpha \mathcal{H}_\alpha$, where $\alpha$ labels irreps of some group $G$...

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### Maximal generalized symmetric groups and the tensor product

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 ...

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### 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 ...

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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 ...

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### 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 ...

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### Partial sum of Weingarten functions over symmetric group

I have a question about partial sums of Weingarten functions. The Weingarten functions are defined as
$$
E_U[U_{i_1,j_1}\dotsm U_{i_k,j_k}U^*_{i'_1,j'_1}\dotsm U^*_{i'_k,j'_k}]=\sum_{\alpha,\beta \in \...

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### Irreducible group representation(algebraic and topological irreducibility)

In page 280 of "C^* algebra" by Dixmier, in the context of group representation, it is written 'We never encounter the concept of algebraic irreducibility except in finite dimensional ...

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### The density of the image of a unitary irrep (a generalization of Burnside's theorem)

I asked the following question on MSE and never got an answer.
I am curious if there are any generalizations of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(...

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### Characterize this subspace of the bounded operators on $ L^2(\mathbb{R}) $

I posted this on MSE a couple months ago and it got three upvotes but no answers or even comments so I decided to cross-post it here:
For every pair $ a,b $ of real numbers define the operator $ U_{a,...

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### Bounding the dimensions of faithful representations of a quotient group

For $G$ a compact Lie group, let $\operatorname{mdfr}(G)$ be the minimum dimension of a faithful complex representation of $G$. Is there a bound on $\operatorname{mdfr}(N(H)/H)$ for $H$ a subgroup of ...

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### 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 ...

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### Zero entropy and the Koopman representation

Let $T$ be a measure preserving bijection of a probability space $(X,\nu)$. Consider the Koopman representation of $\mathbb{Z}$ on $L^2(X,\nu)$ given by $[z.f](x) = f(T^{-z}(x))$. The question is: can ...

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### Smallest dimension for faithful orthogonal representation

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$The compact simple Lie groups $\SO_8(\mathbb{R}) $ and $\SO_9(\mathbb{R}) $ both have rank 4. The group
$$
G=\SU_3 \times \SU_2 \times \...

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### Continuity of left regular representation on space of continuous functions

I am teaching a course on locally compact groups and their representation theory and I am at a point where I would like to introduce continuous representations as generally as possible and provide ...

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### Find unitary transformation between two sets of matrices that represent group generators

I have a set of matrices $A_i$ that represent the generators of a finite group within a certain basis, and $B_i$ represent the same operators in a different basis.
How can I find a unitary ...

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### Generating $K$-types of a $(\mathfrak g,K)$-module for $K$ disconnected

Let $G$ be a real reductive Lie group, let $K$ be a maximal compact subgroup of $G$, and let $V$ be a $(\mathfrak g,K)$-module. For $\sigma\in\widehat{K}$ we denote the $\sigma$-isotypic component of $...

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### Schur positivity of a polynomial

Suppose a polynomial of the form
$$\prod_i^d \sum_j^p x_i^{f_j}$$
clearly symmetric, where $f_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur ...

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241
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### Understanding the regular representation of an LCA group as a 'direct integral'

The reference for what I'm asking is page $107$ from Folland's harmonic analysis.
$G$ is a locally compact abelian group with dual $\hat{G}$. Let $H$ denote the Hilbert space $L^2(G)$.
I'm trying to ...

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### 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 ...

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### Eigenvalues of product of unitaries

Consider $d\times d$ unitary matrices $U, \, V, \, W$ such that
$$
W=UV.
$$
Suppose that the eigenvalues of $U$ and $V$ are $(e^{i\theta_1},\cdots,e^{i\theta_d})$ and $(e^{i\phi_1},\cdots,e^{i\phi_d})$...

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### Haar measure coming from Pontryagin duality v/s Fourier inversion

Not research but advertising this question from mse in case someone wants to answer.
I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...

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### A p-adic analogue of a result due to Kirillov

Let $k$ be a non-Archimedean local field with char$(k)=0$.
Let $N$ be the group of $k-$rational points of a unipotent algebraic group defined over $k$.
It is known that $N$ is a locally compact and ...

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### Induced representations: space of continuous functions on $G$ to a Hilbert space

This question was asked in math stack exchange but received no replies:
https://math.stackexchange.com/questions/4000655/induced-representations-space-of-continuous-functions-on-g-to-a-hilbert-space
...

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### Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation

We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v_0 \in V$ such that $\textrm{Stab}(v_0)= H$. This gives a $...

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### 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 (...

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### 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 ...

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### Unitary representation is strictly continuous

Let $G$ be a compact group and $u: G \to B(H)$ be a strongly continuous unitary representation on the Hilbert space $H$. Then is $u: G \to B(H)$ strictly continuous?
That is, give $B(H)$ the topology ...

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### 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 ...

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### When can an $\mathfrak{S}_n$-equivariant map be extended to an $\textrm{O}(n)$-equivariant map?

The symmetric group $\mathfrak{S}_n$ can be regarded as a subgroup of the orthogonal group $\textrm{O}(n)$ via the permutation matrices. Let $V$ be a finite dimensional $\textrm{O}(n)$-module and $\...

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### 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 ...

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### Realization of limit of discrete series using Dirac operators

I wonder if there is a geometric realization of limit of discrete series in the flavor of Atiyah-Schmid or Parthasarathy realizing discrete series using Dirac operators on G/K. I know you can see ...

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### Kazhdan Property T of semisimple Lie groups

I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M.,
Analogs of Wiener's ergodic theorems for semisimple Lie groups. II.
Duke Math. J. 103 (2000), no. 2, 233–259] (MSN).
I want to ...

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### Examples of groups admitting a proper $1$-cocyle for a bounded representation

A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\...

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### 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}...

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### Parametrization of real-valued SU(N)

I want to construct a $SU(N)$ matrix $V$, with the following property:
All the elements of the first row are given, i.e. $V_{1,j}=a_i$ (with $\sum_i a_i^2=1$)
All matrix elements are real, i.e. $V_{i,...

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### Baker–Campbell–Hausdorff formula for exponential of general Hermitian operators

Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $...

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### Commutation fo a self-adjoint operator with a unitary operator

Let $A$ be a selfadjoint bounded operator on a Hilbert space. Let $M$ be another bounded selfadjoint operator. Let me assume the commutation property
$$
[A, e^{iM}]=0.
\tag 1$$
Does (1) imply that
$$...

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### Average of the maximum matrix element over the Haar measure

Let $U$ be a $d\times d$ unitary matrix, and $U_{i,j}$ be its matrix elements. I am interested in the following quantity
$$\int dU \max_j |U_{1,j}|^2 \ , $$
where $dU$ is the uniform Haar measure over ...

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### Unitary condition

I came across the following while doing some related proof;
It seems easy to prove. $\quad$
We are in ${\mathbb{M}}_n(\mathbb{C})$, $n>1$:
$1$) Given a unitary $n\times n$ matrix $U$, there is ...