Questions tagged [unitary-representations]
The unitary-representations tag has no usage guidance.
194 questions
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Is every finite group a group of "symmetries"?
I was trying to explain finite groups to a non-mathematician, and was falling back on the "they're like symmetries of polyhedra" line. Which made me realize that I didn't know if this was actually ...
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5
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Induction and Coinduction of Representations
I'd like to understand the general framework of induction and coinduction of representations. If G is a finite group and H a subgroup, I know that there is a restriction functor from representations ...
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3
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Is a reductive adelic group a Type I group?
I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is ...
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Unitary representations of SL(2, R)
I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group G being ...
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Good source for representation of GL(n) over finite fields?
I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated.
======== edit =========
My original question was ambiguous. ...
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Regarding Cayley Graphs of Property (T) Groups
A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of ...
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Which groups are the unitary group of a $C^*$-algebra
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
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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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When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$?
A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, ...
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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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Positive definite function zoo
I've asked the following question on math.stackexchange but there has been no response so I'll ask it again here:
A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a ...
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Is there a characterization of free groups in terms of the unitary dual?
If $G$ is a countable discrete group, I'm curious if it is possible to decide whether $G$ is a free group only by looking at properties of $Rep(G)$, the collection of (equivalence classes of) strongly ...
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Representing SU(3) with 3 ropes in 3 dimensions
The short question is: how exactly is SU(3) realized with ropes?
The long question: There is this idea that deformations of a configuration of three infinitely long, flexible ropes that cross each ...
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answer
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Unitary representations of Quantum Groups
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am ...
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Trace Class Functions on locally compact groups
Let $G$ be a locally compact subgroup, $\mu$ a Haar-measure.
For $f \in L^1(G)$, and for $\pi$ a unitary, topology irreducible, representation of $G$ on
an Hilbert space $H_\pi$, it is customary to ...
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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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Groups without property (T) but all finite quotients are expanders
What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G \...
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Characters of permutation groups
Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of
permutations on an $N$-element set that have exactly $m$ cycles (counting
$1$-cycles). Then it is in the literature that the ...
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3
answers
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Topology on the Unitary Dual
Suppose I have a locally compact topological group G. The unitary dual of G is the set of equivalence classes of irreducible unitary representations of G. Now, it seems to me that the sensible way of ...
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Quantum Cellular Automata on Riemannian manifolds and geometric group theory
We try to motivate our question. We have a certain logical/operational structure that has an
emergent physical interpretation. We are giving this structure a geometric setting via
quasi-isometries. ...
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Unitary representations of the ax+b group: an accessible presentation
The "ax+b group" is the group of affine transformations of $\mathbb R$. It is a locally compact non unimodular group.
Its space of irreducible, continuous unitary representations has been described ...
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Induced representations of topological groups
Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups)
If $G$ is a group ...
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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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Restriction of irreducible unitary representation to normal subgroup of finite index
Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...
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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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434
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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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652
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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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Tensor products of unitary irreducible representations of $SU(2,2)$
What is known about irreducible decomposition of tensor products of (infinite-dimensional) unitary irreducible representations of $SU(2,2)$ (or, more generally, simple groups of split rank greater ...
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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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Can monomial representations induced from nonmonomial representations?
Let $H$ be a subgroup of $G$. Let $\rho$ be an irr representation of $G$ induced from an irr representation $\theta$ of $H$. It is well known that $\rho$ is monomial if $\theta$ is monomial. Is it ...
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Unbounded representations of groups
Let $H$ be a Hilbert space and $G$ be a finitely generated group. Let $\pi:G\rightarrow GL(H)$ be a representation.
A map $c:G\rightarrow H$ is called cocycle if $c(gh)=π(g)c(h)+c(g)$ for all $g,h$ ...
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Faithful representation of the projective unitary group with the lowest dimension?
What is the lowest dimension of a faithful ordinary representation (as compared with projective representation) of the projective unitary group $\rm{PU}(d)$? Is it $d^2-1$?
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Decomposing an arbitrary unitary representation of a connected nilpotent Lie group in terms of its irreps
For a locally compact (Hausdorff) abelian group $G$ we have following theorem (see e.g. Folland):
"For every (strongly continuous) unitary representation $(\pi,\mathcal{H_{\pi}})$ of $G$, there ...
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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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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 ...
user167952
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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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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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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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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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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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What is the difference between a primary representation and a irreducible representation?
I am currently reading some of Mackey's work on unitary representation.
Given a locally compact group $G$ and a unitary representation $\pi : G\rightarrow U(H)$. As far as I understood it, the ...
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Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?
I'm asking a question about Lie group representation.
Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...
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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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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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Unitary irreps of the Poincare group in dimension <4
It is well-known that long ago, Wigner classified the unitary irreducible representations of the Poincare group in dimension 4.
I am looking for a convenient reference describing all unitary ...
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A sufficient condition (or not) for positive semidefiniteness of a matrix?
Let $A\in M_{n}(\mathbb{C})$ be a Hermitian matrix. If for all $z_1,...,z_n\in\mathbb{C}$, $$\sum_{i,j=1}^{n}A_{ij}z_{i}\overline{z_{j}}\ge 0$$ then A is positive semidefinite.
I do not think the ...
- 7,057
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Arthur's refinement of parameters for unitary automorphic representations
In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...
- 1,073
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Representation over matrices $A_i^3=I$, $A_0A_1^\dagger+A_1A_2^\dagger+A_2A_0^\dagger=0$, $A_0^\dagger A_1+A_1^\dagger A_2+A_2^\dagger A_0=0$
I would like to know what all the possible finite-dimensional representations of the following relations are.
$$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^\...
- 583
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references for faithful orthogonal (or unitary) representation of symmetric groups
Let $S_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1).
(1). There does not exist any faithful orthogonal representation
$$
S_n\...
- 543
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Spherical functions for sl(2,Q_p)
I kindly would like to ask you the following- I am refering to page
175 in the Book by Gelfand, Graev, Shapiro, etc, on "Automorphic forms
..."
My question to which I would kindly ask you to answer ...
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