Questions tagged [unitary-representations]
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194 questions
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relationships between $AA^T$ and $[(I-A)(I-A)^T]^{-1}$ with $A$ being strictly lower triangular
I have a matrix $A$ which is strictly lower triangular. Now, I am trying to find some general statements/relationships of following matrices $U,D,V,K$ defined as:
$AA^T=VKV^H$,
$[(I-A)(I-A)^T]^{-1}=...
- 21
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5
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2k
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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. ...
- 297
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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-...
- 493
3
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89
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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 $...
- 933
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187
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Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two
What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
- 1,702
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81
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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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0
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74
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Operators associated with unitary representations of nilpotent Lie group
Let $G$ be a nilpotent Lie Group, and $\pi:G\to B(\mathcal H)$ be an irreducible unitary representation on the Hilbert space $\mathcal H$. One can use the Bochner integral to define a linear map as ...
- 106
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1
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823
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Principal series representations of $SL(2,\mathbb{R})$: introductory textbooks [duplicate]
I am interested in introductory books/papers/reports about the (unitary) representation theory of $SL(2,\mathbb{R})$, with particular emphasis on the principal series representations. My background: I ...
- 329
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0
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77
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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 $\...
- 5,407
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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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2
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597
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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, ...
- 2,851
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1
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802
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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 $\...
- 333
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2
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333
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Unitary representations of SO(1,4) and SO(2,3)
Where can I find details about the irreducible unitary representations of SO(1,4) and SO(2,3)?
- 3,873
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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 ...
- 3,873
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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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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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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 $...
- 2,928
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232
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Kirillov orbit Method for Complex nilpotent groups
Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...
- 2,508
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2
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779
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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 ...
- 4,679
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0
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138
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Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series
Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
- 1,702
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445
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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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2
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537
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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 \...
- 4,432
5
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1
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924
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On the definition of matrix coefficient
As far as I have known, for irreducible admissible representation $\pi$ of $p$-adic group $G$, the matrix coefficient is defined as follows:
For $v\in \pi$ and $w \in \pi ^\vee$, the contragredient ...
- 1,019
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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 ...
- 5,565
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836
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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})$ ...
- 419
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144
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Unitarizability of group representations
Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...
- 21.8k
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300
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The representation theory for the fake Heisenberg groups over non-perfect local field
Let $K$ be a local field of characteristic $p$, where $p$ is a prime number greater than 2. In particular, $(x+y)^p=x^p+y^p$ for $x,y\in K$.
The fake Heisenberg group is defined to be
$$
G=\{\begin{...
- 1,702
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0
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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 ...
6
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1
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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 ...
- 3,873
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139
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Representation of finite group
Let $\Gamma$ be a finite index subgroup of $SL_2(\mathbb{Z})$. Let $\pi$ be a natural covering from $\Gamma\backslash\mathbb{H}$ to $SL_2(\mathbb{Z})\backslash\mathbb{H}$. Denote by $\text{Deck}(\pi)$ ...
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2
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535
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Is anything known about the eigenspectrum of the regular representation of the permutation group?
I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known?
The ...
- 1,893
2
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2
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68
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Animating a unitary transform [closed]
For the purpose of showing how (quantum) unitary operators behave as a computer animation, I would like to create a function $A_U(t)$ of some (complex) unitary matrix $U$, such that $A_U(t)$ is ...
- 121
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Decomposing a reducible representation of the unitary group
Consider the representation $L_U$ of the unitary group $U(n)$ on $L(\mathbb{C}^n)$ where $L_U$: $L(\mathbb{C}^n) \rightarrow L(\mathbb{C}^n)$ is a linear operator that $L_U M=U M U^{\dagger} $, $\...
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Canonical representation of $\operatorname{SL}(2,\mathbb{R})$ on $L^2(\mathbb{R}^2)$
As a unimodular subgroup of the group of automorphisms of $\mathbb{R}^2$, $\operatorname{SL}(2,\mathbb{R})$ can be represented as a subgroup of $\mathcal{U}(L^2(\mathbb{R}^2))$ (the group of unitary ...
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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 ...
0
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1
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558
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Schur's lemma for antiunitary operators on complex Hilbert spaces
Suppose to have a linear irreducible unitary representation $\rho:G\rightarrow U(H)$ on a complex Hilbert space $H$ with $G$ a generic group. Let $A$ be an $\textit{anti}$-linear operator such that
$$...
- 275
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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, ...
- 647
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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 ...
- 541
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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 ...
- 647
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2
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244
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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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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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193
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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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1
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Analytic criteria for the support of the Plancherel measure for SL(2,Qp), spherical functions
Is there an analytic criteria to determine the support of the Plancherel measure for SL(2,Qp). At least for unitary spherical representations
- 193
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Do the irreducible unitary representations of a locally compact group form a separating set for the Radon measures on the group?
Let $\mu$ and $\nu$ be two Radon measures on a locally compact group $G$. For every irreducible unitary representation $\pi$ of $G$ and vectors $u$ and $v$ from the corresponding Hilbert space $H_\pi$ ...
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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 ...
- 193
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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 $G.$
...
- 2,689
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About the group generated by one diagonal unitary
Suppose $D=diag\{\alpha_1,\alpha_2,...\alpha_n\}$ is a diagonal unitary, which means that |\alpha_i|=1 for all $i$. We know that $\alpha_i$ is not unit root and so is $\alpha_i/\alpha_j$ for $i\neq j$....
- 1,503
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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 $GL(2, \mathbb{C})$, ...
- 11.2k
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1
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549
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Supercuspidal with Iwahori fixed vector
Let $F$ be a local field. Is there a reference for the following fact:
No supercuspidal representation of $GL_2(F)$ has an Iwahori-fixed vector?
I have a proof, by I'd prefer a reference, because ...
- 11.2k
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Isometric representation semisimple?
The first lemma on p.35 of these notes states that unitary representations are semisimple. Could the same be said of isometries if the space doesn't have an inner product? This topic notes that the ...
- 1,049