# Questions tagged [unitary-representations]

The unitary-representations tag has no usage guidance.

183
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### *-Representation of a discrete group as a sum of cyclic representations

A cyclic representation on a Hilbert space is one where there exists a vector such that, under the action of the representation, this vector spans the entire Hilbert space.
It is known that every *-...

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### Question on two types of Frobenius theorem in $p$-adic groups

Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a
standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $\text{Jac}$ be the ...

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264
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### Book on Hilbert spaces, including non-separable

I am looking for a book that develops the theory of Hilbert spaces, including the spectral theorems and unitary representations, but includes non-separable Hilbert spaces in the main exposition. Any ...

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### Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...

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### About normal states in abstract von Neumann algebras

In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16
but this was state only for concrete von Neumann algebras (because ...

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### A question on projective unitary representation of a Lie group

$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...

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### Restricting unitary irreducible representations of the Poincaré group

The Poincaré group is the isometry group of Minkowski spacetime and every point in Minkowski spacetime is stabilised by a subgroup of the Poincaré group isomorphic to the Lorentz group. Let us fixed ...

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114
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### How to build a representation of the diffeomorphism group of $U(n)$?

Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n)...

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### Can Rep(G) tell us whether G is discrete?

Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations.
The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...

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### Tempered representations and unramified principal series

For $V$ a tempered representation of connected reductive group over a local field of characteristic zero. I want to show that for an Iwahori subgroup $B$, the set of fixed points $V^B\neq 0$, thereby ...

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### Two definitions of intertwining operators and Harish-Chandra's Plancherel measure

I guess this question is a well-known fact to experts, but I didn't find any explicit explanation in the literature.
So let $F$ be a $p$-adic field. (There're parallel definitions and results in the ...

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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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### 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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### 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 $\...