Questions tagged [unitary-representations]
The unitary-representations tag has no usage guidance.
81 questions with no upvoted or accepted answers
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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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386
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Does every commutative $*$-algebra of operators on a prehilbert space have a character?
My question can be equivalently stated as follows:
Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of linear ...
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299
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when do norm-continuous unitary representations separate points of a group?
Recently I found in the web a discussion on the following question:
...
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509
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Are square-integrable representations of a general locally compact group tempered
Let $G$ be a locally compact unimodular group with center $Z$ and let $\omega$ be a unitary character of $Z$. To fix the discussion, an irreducible unitary representation $\pi$ with central character $...
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602
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Unique maximal ideal in group C*-algebras
Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$
...
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6
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240
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Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces
I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...
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160
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Proof that the second Borel cohomology group of $(\mathbb R, +)$ is trivial
Does anyone have a reference for a fairly direct proof that the second Borel cohomology group for $(\mathbb R, +)$ (with the trivial action on the circle group) is trivial? The motivation is to show ...
6
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502
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Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep
Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...
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5
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292
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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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5
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167
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Plancherel formula for $L^2(G/N)$
Let $G$ be a connected real semisimple or reductive Lie group. Let $TA$ be a Cartan subgroup, where $T$ is compact and $A$ is split. Let $MA$ be the centralizer of $A$ in $G$, and let $N$ be the ...
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269
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non unitary representations
A paper by M.L. Whippman in Rep. Math. Phys. 5 (1974), 81, mentions at the bottom of the second page ''the possible occurrence of non-unitary representations that arise when reducing a direct product ...
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5
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321
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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, ...
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4
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143
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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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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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73
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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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128
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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 $U(n)$ ...
user340793
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124
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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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89
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Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?
This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...
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119
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Number of unitary representations of a Kazhdan group
It was proved by de la Harpe, Robertson, and Valette that for a discrete group $\Gamma$ with Kazhdan's property (T), there is a constant $c$ so that the number of irreducible unitary representations ...
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How to find the unitary matrices in this exponential matrix representation
In the following post
Representing a product of matrix exponentials as the exponential of a sum
there is a statement regarding the result of the multiplication of two matrix exponentials:
if $A$ and $...
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3
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109
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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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3
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40
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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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269
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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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3
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235
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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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178
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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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234
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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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99
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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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3
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106
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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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255
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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 $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...
- 1,759
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Unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$?
The real motion group of $\mathbb R^2$, $M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well
known fact is that the unitary dual $\hat{G}$, of $G$ ...
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141
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Existence of a unique cyclic and separating vector in a *-representation
I'm interested in knowing the requirements for a $*$-representation, $\pi_{\omega}$, of a C*-algebra, $\mathbb{C}(\mathcal{G})$, (or equivalently the requirements for the unitary representation, $U_{\...
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the moment set of unitary representation of lie groups, analogue in the p-adic case
Let $G$ be a real Lie group with Lie algebra $\mathfrak{g}$ and $\pi$ a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_{\pi}$. Note $\mathcal{H}_{\pi}^{\infty}$ the space of $...
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The relation of the local principal representations of $U(2)$ and $GL(2)$
Let $E/F$ be a quadratic extension of number fields and $v$ is a non-archimedean place of $F$.
Let $G=U(2)(F_v)$ be the $F_v$-points of the 2-dimension unitary group associated to $E_v/F_v$ and $B$, $...
- 1,759
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Two Hilbert $G$-bundles are isomorphic iff the representations of the little group are equivalent
While reading Mackey's "Unitary Group Representations in Physics, Probability and Number Theory", at page 66 I encountered the following statement (my reformulation):
if $\mathcal K$ and $\...
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3
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237
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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 ...
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Method to Generate Random Mutually Orthogonal Unitary Matrices
The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
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Character of continuous series representation of GL(2)
It is wellknown that the character of an irreducible, unitary representation of $GL(n,\mathbb{C})$ uniquely determines the isomorphism classes. I fail to construct a function for $GL(2, \mathbb{C})$, ...
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Do local L-functions/epsilon factors vary continuously with the Fell topology?
Edit due to the comment.
Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable.
Given a sequence of irreducible unitary representations $(\pi_n)$ of ...
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70
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Representations of unitary group on spaces of differential forms
This is a question on certain irreducible real representations of the unitary group. My main reference is Salamon's book "Riemannian geometry and holonomy groups".
The unitary group $\mathrm ...
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Ordering of norms and the Shapovalov form on highest weight modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and let $\mathfrak{U}(\mathfrak{g})$ be its universal enveloping algebra. Fix a Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g}$, and ...
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118
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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-...
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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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72
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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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155
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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 ...
2
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141
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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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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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80
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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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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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