Questions tagged [unitary-representations]

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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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2 votes
0 answers
145 views

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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1 vote
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66 views

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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6 votes
1 answer
193 views

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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1 vote
0 answers
81 views

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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9 votes
1 answer
394 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 ...
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3 votes
1 answer
88 views

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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2 votes
1 answer
175 views

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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0 votes
0 answers
93 views

The topological Grothendieck group of a mixed category

I have recently become interested in the notion of mixed categories, as well as the topological Grothendieck group of their derived categories. I am still very new to the field. For that, I would like ...
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1 vote
1 answer
172 views

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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  • 39
4 votes
1 answer
457 views

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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3 votes
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33 views

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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4 votes
2 answers
195 views

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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1 vote
1 answer
195 views

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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7 votes
1 answer
269 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 ...
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3 votes
1 answer
121 views

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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1 answer
341 views

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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1 vote
0 answers
133 views

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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1 vote
1 answer
101 views

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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  • 113
4 votes
1 answer
115 views

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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  • 43
9 votes
1 answer
183 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 (...
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4 votes
0 answers
176 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 ...
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7 votes
1 answer
133 views

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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2 votes
0 answers
122 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 ...
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6 votes
3 answers
371 views

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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  • 2,277
4 votes
1 answer
226 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 ...
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2 votes
0 answers
48 views

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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  • 802
3 votes
0 answers
220 views

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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1 vote
0 answers
77 views

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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4 votes
0 answers
112 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}...
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  • 178
5 votes
3 answers
327 views

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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3 votes
0 answers
128 views

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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1 vote
0 answers
72 views

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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8 votes
2 answers
491 views

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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1 vote
1 answer
266 views

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 ...
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  • 418
4 votes
1 answer
222 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 $\ell^\infty (G) \rtimes_{\mathrm{red}}G$. ...
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7 votes
1 answer
328 views

K-type in discrete series representation

The following result seems well known. Let $G$ be a reductive Lie group with a maximal compact subgroup $K$. If $\mu$ is an irreducible unitary representation of $K$, then there exist only finitely ...
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  • 707
2 votes
1 answer
114 views

Finding all unitary representations of the connected Poincaré group

I am studying representation theory of Lie groups and its combination to theoretical physics, and I am concerned about the following. Is there an exhaustive way to find all unitary representations of ...
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1 vote
0 answers
101 views

Irreducible unitary representations of discrete abelian groups

It seems to me that the statement below should be true but I would like to double-check. Statement: Let $H$ be a (separable) complex Hilbert space and consider its associated unitary group $U(H)$ ...
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3 votes
0 answers
112 views

Question about regular representation of compact group

I first define the setting for my question. Let $G$ be a compact group with probability Haar measure $\mu_G$. Denote by $\lambda$ the left regular representation on $L^2(G)$ defined for $f \in L^2(G)$ ...
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3 votes
0 answers
197 views

Is the character of the adjoint representation of $\operatorname{SU}(n)$ non-vanishing on regular points of a maximal torus?

Are the maximal and minimal values of the character of the adjoint representation of $\operatorname{SU}(n)$ restricted to a maximal torus known? Can such a character vanish at some regular point of a ...
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  • 3,594
6 votes
2 answers
293 views

Representation over matrices $A_i^3=I$, $A_0A_1^\dagger+A_1A_2^\dagger+A_2A_0^\dagger=0$, $A_0^\dagger A_1+A_1^\dagger A_2+A_2^\dagger A_0=0$

I would like to know what all the possible finite-dimensional representations of the following relations are. $$A_0^3 = A_1^3 = A_2^3 = I \tag{1}$$ $$A_0 A_1^\dagger + A_1 A_2^\dagger + A_2 A_0^\...
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  • 543
1 vote
0 answers
54 views

Weyl theorem for non specified primitive root of unity

Let $\omega=e^{2i \pi/p}$. Weyl theorems give all representations of matrix algebra span by $A,B$ such that either $AB=\omega BA, A^p=B^p=I$, or $(k,l)\mapsto A^kB^l$ is a irreducible ...
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  • 543
3 votes
0 answers
95 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 ...
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  • 543
5 votes
2 answers
299 views

Definition of unitary representation of $\mathbf G(\mathbb A_k)$

Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ ...
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  • 5,536
3 votes
1 answer
292 views

tensor product of massless Poincare representations

Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles? Massless ...
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3 votes
0 answers
71 views

Restriction that contains a trivial representation

Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$ such that $(G,G')$ is a symmetric pair. If $\pi$ is an infinitely dimensional unitary representation of $G$, ...
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  • 707
10 votes
1 answer
763 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,\...
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5 votes
1 answer
1k views

Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices

Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that $$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$ But suppose I ...
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3 votes
0 answers
171 views

Some basic question on the parabolic induction

I would like to ask some basic question about parabolic induction. Let $F$ be a local field and $G=GL_n(F)$ and $P=MN$ its parabolic subgroup whose Levis subgroup $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...
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