Questions tagged [unitary-representations]
The unitary-representations tag has no summary.
203 questions
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Has the hydrogen representation as minimal K-type the trivial representation?
The irreducible representations of the compact group $SO(4,R)$ are classified by pairs of so called “spin-quantum numbers” $(j_1, j_2 )$ with $j_1, j_2$ non-negative integers or half-integers.
The ...
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Non-Unitary comodules of Hopf *-algebras
Let $H$ be a Hopf algebra endowed with a $*$-structure, making it a Hopf $*$-algebra. Take a finite-dimensional $H$-comodule $(V,\Delta_V)$ and choose a basis $e_i$. In terms of this basis we have ...
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Diagonalization of a monomial matrix with constraints
Definitions
Let $G$ be a finite subgroup of $U(n)$ and let $\mathcal{D} \subset U(n)$ denote the group of $n\times n$ diagonal, unitary matrices. We'll say that a matrix $T$ is diagonalizable over $G$ ...
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Are unitary representations of compact Lie groups Bochner integrable?
Let $G$ be a compact Lie group endowed with the bi-invariant Haar measure $\mu$ of total mass $1$. Let $\Phi \colon G \to \mathcal{B}(H)$ be a unitary representation of $G$ on a Hilbert space $H$, ...
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Reducing group representations with Dixon's method
I am trying to find allirreducible representations of a finite group using Dixon's method ("Computing Irreducible Representations of Groups"). The matrix elements need to be in algebraic ...
- 340
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129
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Examples of Harish-Chandra Schwartz functions coming from tempered representations
Let $F$ be a local field and (for simplicity) $G$ a semisimple group split over $F$. Then for in both the archimedean and non-archimedean cases, one has the space $\mathcal{C}(G)$ of Harish-Chandra ...
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How can I define an invariant inner product on a Schur module?
As the title. In general, given a partition or say, a Young diagram $\lambda$ and a linear space $V$ of dimension $n$, we can construct a Schur module corresponding to the irreducible $GL(n)$-...
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Non-tempered unramified spectrum of semisimple group
$\newcommand{\nt}{\mathrm{nt}}$Let $G$ denote a semisimple linear algebraic group over a local field $F$.
Fix a maximal compact subgroup $K$ and write $\widehat G$ for the unitary dual of $G$, ...
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154
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Representation theory applied to abstract harmonic analysis
Let $G$ be a compact group and $(\pi_1,H_1),(\pi_2,H_2)$ to unitary and irreducible representation which are not unitarily equivalent.
My question is the following: Is there some way two guarantee the ...
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Why the unitary dual of a locally compact group is a set?
If $G$ is a locally compact group, the definition of unitary dual of $G$ is the set of all equivalence classes of irreducible unitary representations of $G$, denoted by $\widehat{G}$. But why this is ...
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Invariant theory for unitary groups $\mathcal{U}(n)$
I'm trying to understand the invariant theory of the unitary groups $\mathcal{U}(n)$ on tensor powers of their standard representations $V^{\otimes p} \otimes (V^*)^{\otimes q}$. Let $\mathcal{U}(n)$ ...
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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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Unitary representations of the symmetric group over finite fields
I am interested in understanding the unitary representations of the symmetric group over $\mathbb{F}_{q^2}$. In general, some comments here are relevant
Unitary representations of finite groups over ...
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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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K-types of a representation of the minimal Gelfand-Kirillov dimension
Let $G$ be a noncompact real simple Lie group not of Hermitian type, and $\mathfrak{g}_0$ its Lie algebra. Fix a maximal compact subgroup $K$ in $G$ with its Lie algebra $\mathfrak{k}_0$. Write $\...
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Linear algebraic group, absolute root system, computing roots
Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
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Paper request: Graev's classification of SU(2,2) irreducible unitary representations
I am interested in Graev's paper in "M. L. Graev:Dokl. Akad. Nauk SSSR,98, 517 (1954); Amer. Math. Soc. Transl.,66, 1 (1968)." in which the irreducible unitary representations of SU(2,2) are ...
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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 ...
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170
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Exponentiating a representation of a semi-simple Lie algebra
I consider a representation of a semi-simple Lie algebra $\mathfrak{g}$ (specifically, the symplectic and orthogonal Lie algebras $\mathfrak{sp}(2N)$ and $\mathfrak{so}(2N)$) as anti-Hermitian ...
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129
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Sampling orthogonal matrices from Haar-random unitary group
I would like to know the probability of sampling orthogonal matrices $O \in O(d)$ from Haar-random unitary group $U(d)$. The probability may be close to zero since orthogonal matrices are "sparse&...
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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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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}:\...
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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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3
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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 ...
- 141
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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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136
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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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166
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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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158
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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 $\...
user473423
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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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2
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229
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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)$ ...
user340793
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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 ...
- 123
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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 ...
- 400
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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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1
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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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