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Consider an infinite homogeneous tree $T_n$ of valence $n+1$, its group $Aut(T_n)$ of automorphisms, and the group $Hie(T_n)$ of its spheromorphisms (hierarchomorphisms), i.~e., the group of homeomorphisms of the boundary of $T_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup $Aut(T_n)$ is spherical in $Hie(T_n)$, i.~e., any irreducible unitary representation of $Hie(T_n)$ contains at most one $Aut(T_n)$-fixed vector. We present a combinatorial description of the space of double cosets of $Hie(T_n)$ with respect to $Aut(T_n)$ and construct a 'new' family of spherical representations of $Hie(T_n)$. We also show that the Thompson group has $PSL(2,\mathbb{Z})$-spherical unitary, Comment: 18pages, 8 figures

El acceso a los instrumentos de comunicación digital existentes en la empresa, como el correo electrónico o la intranet, resulta esencial para que los órganos de representación unitaria y sindical puedan desarrollar adecuadamente sus funciones. Nuestro marco legal nunca ha garantizado tal acceso pero, a través de la STC 281/2005, de 7 de noviembre, se reconoció, con fundamento en el contenido esencial de la libertad sindical, un derecho de uso sobre esos instrumentos cuando se hayan implantado en el proceso productivo y sean adecuados para el desarrollo de la actividad representativa. No obstante, los límites impuestos por la propia jurisprudencia constitucional a ese derecho de uso y la forma como han sido aplicados por la jurisdicción ordinaria, han supuesto en la práctica un obstáculo importante, a menudo insuperable. Resulta imprescindible abordar la actualización del marco legal sobre los instrumentos de comunicación de las representaciones laborales, para garantizarles, desde la norma heterónoma y con carácter general, el derecho a utilizar los instrumentos empresariales de comunicación digital en el ejercicio de sus funciones. L'accés a les eines de comunicació digital existents a l'empresa, com el correu electrònic o la intranet, són essencials per tal que els òrgans de representació unitària i sindical puguin desenvolupar les seves funcions de forma adequada. El nostre marc legal no ha garantit mai aquest accés, però a través de la STC 281/2005, de 7 de novembre, es va reconèixer, amb fonament al contingut essencial de la llibertat sindical, un dret d'ús en relació a aquestes eines quan s'hagin implantat al procés productiu i siguin aptes pel desenvolupament de l'activitat representativa. No obstant, els límits imposats per la pròpia jurisprudència constitucional a aquest dret d'ús i la forma com els ha aplicat la jurisdicció ordinària, han generat a la pràctica un obstacle important, sovint insuperable. És imprescindible abordar l'actualització del marc legal sobre les eines de comunicació de les representacions laborals, per tal d'assegurar, des de la norma heterònoma i amb caràcter general, que puguin utilitzar els canals empresarials de comunicació digital per a l'exercici de les seves funcions. Access to existing digital communication tools in the company, such as e-mail or intranet, is essential for the union and trade union representation bodies to be able to carry out their functions properly. Our legal framework has never guaranteed such access, but through the STC 281/2005, of November 7th, it was recognised, based on the essential content of trade union freedom, the right to use these instruments when they have been implemented in the production process and are appropriate for the development of representative activity. However, the limits imposed by constitutional jurisprudence itself on this right, and the way in which they have been applied by the ordinary courts have, in practice, represented a major, often insurmountable, obstacle. It is essential to update the legal framework of the communication tools used by workers' representatives, in order to guarantee them, from the heteronomous standard and in general, the right to use the company's digital communication tools in the exercise of their functions.

We show that the groups of finite energy loops and paths (that is, those of Sobolev class $H^1$) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation $\pi$ of such a group (which we call skew-amenable) has a conjugation-invariant state on $B({\mathcal H}_{\pi})$., Comment: 18 pp., latex with elsevier macros. A typo in the statement of Lemma 9 was corrected, and Lemma 11 was stated more accurately

In this paper, we study the quasi-regular and the irreducible unitary representation of affine Lie group of dimension two. First, we prove a sharpening of Fuhr’s work of Fourier transform of quasi-regular representation of . The second, in such the representation of affine Lie group is square-integrable then we compute its Duflo-Moore operator instead of using Fourier transform as in F hr’s work.

In this paper, we study a harmonic analysis of a Lie group of a real filiform Lie algebra of dimension 5. Particularly, we study its irreducible unitary representation (IUR) and contruct this IUR corresponds to its coadjoint orbits through coadjoint actions of its group to its dual space. Using induced representation of a 1-dimensional representation of its subgroup we obtain its IUR of its Lie group

We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $\Gamma$ to the setting of a general unitary representation $\pi: \Gamma \to B(\mathcal H_\pi)$. This space, which we call the "Furstenberg-Hamana boundary" of the pair $(\Gamma, \pi)$, is a $\Gamma$-invariant subspace of $B(\mathcal H_\pi)$ that carries a canonical $C^*$-algebra structure. In many natural cases, including when $\pi$ is a quasi-regular representation, the Furstenberg-Hamana boundary of $\pi$ is commutative, but can be non-commutative in general. We study various properties of this boundary, and give some applications., Comment: latter part of Lemma 7.3 removed

As a starting point we assume to have a continuous frame in a Hilbert space with respect to a measure space. This frame inherits a unitary structure from a unitary representation of a locally compact abelian group in the Hilbert space. In this setting we state a continuous sampling result for the range space of the associated analysis frame operator. The data sampling are functions also defined by using the underlying unitary structure. The result is illustrated by using continuous frames in Paley–Wiener and shift-invariant spaces generated by translates of fixed functions. A sampling strategy working only for discrete abelian groups is also discussed.

We describe how to introduce dynamics for the holographic states and codes introduced by Pastawski, Yoshida, Harlow and Preskill. This task requires the definition of a continuous limit of the kinematical Hilbert space which we argue may be achieved via the semicontinuous limit of Jones. Dynamics is then introduced by building a unitary representation of a group known as Thompson's group T, which is closely related to the conformal group in 1+1 dimensions. The bulk Hilbert space is realised as a special subspace of the semicontinuous limit Hilbert space spanned by a class of distinguished states which can be assigned a discrete bulk geometry. The analogue of the group of large bulk diffeomorphisms is given by a unitary representation of the Ptolemy group Pt, on the bulk Hilbert space thus realising a toy model of the AdS/CFT correspondence which we call the Pt/T correspondence., 40 pages (revised version submitted to journal). See video of related talk: https://www.youtube.com/watch?v=xc2KIa2LDFo

We consider a task in which classical information is encoded into a quantum system by an operation restricted by symmetry. The maximum amount of classical information that can be encoded under this restriction, namely the symmetry-restricted classical information capacity, depends on the initial state of the system. Our focus is on whether the capacity of an asymmetric state can be strictly larger than that of any symmetric states, whereas the latter is a strictly positive constant. That is, we ask whether an analog of superdense coding is implementable in the resource theory of asymmetry. We prove that superdense coding is implementable if and only if the unitary representation of the symmetry is non-Abelian and reducible. Thereby we provide an information theoretical classification of symmetries of quantum systems., Comment: 7 pages, 1 figure

We will introduce the notion of inductive limits of compact quantum groups as $W^*$-bialgebras equipped with some additional structures. We also formulate their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantized characters associated with a given inductive system of compact quantum groups. As a byproduct, we will give an explicit representation-theoretic interpretation to some transformations that play an important role in the analysis of $q$-central probability measures on the paths in the Gelfand-Tsetlin graph., 17 pages, second version (added an appendix)

Continuing the study of the Moyal Higher Spin Yang–Mills theory started in our previous paper we provide a detailed discussion of matter coupling and the corresponding tree-level amplitudes. We also start the investigation of the spectrum by expanding the master fields in terms of ordinary spacetime fields. We note that the spectrum can be consistent with unitarity while still preserving Lorentz covariance, albeit not in the usual way, but by employing an infinite-dimensional unitary representation of the Lorentz group.

The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.

Let $G$ be a discrete countable group, and let $\Gamma$ be an almost normal subgroup. In this paper we investigate the classification of (projective) unitary representations $\pi$ of $G$ into the unitary group of the Hilbert space $l^2(\Gamma)$ that extend the left regular representation of $\Gamma$. Representations with this property are obtained by restricting to $G$ square integrable representations of a larger semisimple Lie group $\bar G$, containing $G$ as dense subgroup and such that $\Gamma$ is a lattice in $\bar G$. This type of unitary representations of of $G$ appear in the study of automorphic forms. We prove that the Ramanujan-Petersson problem regarding the action of the Hecke algebra on the Hilbert space of $\Gamma$-invariant vectors for the unitary representation $\pi\otimes \bar\pi$ is an intrinsic problem on the outer automorphism group of the von Neumann algebra $\mathcal L(G \rtimes L^{\infty}(\mathcal G,\mu))$, where $\mathcal G$ is the Schlichting completion of $G$ and $\mu $ is the canonical Haar measure on $\mathcal G$.

We examine some connections among phase-retrievable (not necessarily self-adjoint) operator-valued frames, projective group representation frames and representations of quantum channels. We first present some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces, which generalize the known results for vector-valued frames, fusion frames and frames of Hermitian matrices. For an irreducible projective unitary representation of a finite group, the image system is automatically phase-retrievable and, moreover, it is a point-wisely tight operator-valued frames. We generalize this notion to more general operator-valued frames, and prove that point-wise tight operator-valued frames are exactly the ones that are right equivalent to operator-valued tight frames. For an operator system that represent a quantum channel, we show that phase-retrievability of the system is independent of the choices of the representations of the quantum channel.

Let $G=NH$ be a Lie group where $N,H$ are closed connected subgroups of $G,$ and $N$ is an exponential solvable Lie group which is normal in $G.$ Suppose furthermore that $N$ admits a unitary character $\chi_{\lambda}$ corresponding to a linear functional $\lambda$ of its Lie algebra. We assume that the map $h\mapsto Ad\left( h^{-1}\right) ^{\ast}\lambda$ defines an immersion at the identity of $H$. Fixing a Haar measure on $H,$ we consider the unitary representation $\pi$ of $G$ obtained by inducing $\chi_{\lambda}.$ This representation which is realized as acting in $L^{2}\left( H,d\mu_{H}\right) $ is generally not irreducible, and we do not assume that it satisfies any integrability condition. One of our main results establishes the existence of a countable set $\Gamma\subset G$ and a function $\mathbf{f}\in L^{2}\left( H,d\mu_{H}\right) $ which is compactly supported and bounded such that $\left\{ \pi\left( \gamma\right) \mathbf{f}:\gamma\in\Gamma\right\} $ is a frame. Additionally, we prove that $\mathbf{f}$ can be constructed to be continuous. In fact, $\mathbf{f}$ can be taken to be as smooth as desired. Our findings extend the work started in \cite{oussa2018frames} to the more general case where $H$ is any connected Lie group. We also solve a problem left open in \cite{oussa2018frames}. Precisely, we prove that in the case where $H$ is an exponential solvable group, there exist a continuous (or smooth) function $\mathbf{f}$ and a countable set $\Gamma$ such that $\left\{ \pi\left( \gamma\right) \mathbf{f}:\gamma\in\Gamma\right\} $ is a Parseval frame. Since the concept of well-localized frames is central to time-frequency analysis, wavelet, shearlet and generalized shearlet theories, our results are relevant to these topics and our approach leads to new constructions which bear potential for applications., Comment: 44 pages

We describe a class of irreducible non-equivalent unitary representations of the braid group B 3 in every dimension n ≥ 6 which depends continuously on n 2 / 6 + 1 real parameters. We show that the upper bound on the number of the parameters of which the class of irreducible non-equivalent unitary representations of B 3 depends smoothly is equal to n 2 / 4 + 2 . The proof is achieved by a construction of such a class. We also prove that the tensor product of the Burau unitarisable representation of B 4 and the irreducible unitary representation of B 4 that coincide on commuting standard generators always forms irreducible unitary representations for the braid group B 4 . This gives a new class of unitary representations for the braid group B 4 in 3n dimensions.

The C ⁎ -subalgebra B of all bounded linear operators on the space L 2 ( R ) , which is generated by all multiplication operators by piecewise slowly oscillating functions, by all convolution operators with piecewise slowly oscillating symbols and by the range of a unitary representation of the group of all affine mappings on R , is studied. A faithful representation of the quotient C ⁎ -algebra B π = B / K in a Hilbert space, where K is the ideal of compact operators on the space L 2 ( R ) , is constructed by applying an appropriate spectral measure decompositions, a local-trajectory method and the Fredholm symbol calculus for the C ⁎ -algebra of convolution type operators without shifts. This gives a Fredholm symbol calculus for the C ⁎ -algebra B and a Fredholm criterion for the operators B ∈ B .

Let $G$ be an infinite-dimensional real classical group containing the complete unitary group (or complete orthogonal group) as a subgroup. Then $G$ generates a category of double cosets (train) and any unitary representation of $G$ can be canonically extended to the train. We prove a technical lemma about the complete group $GL$ of infinite $p$-adic matrices with integer coefficients, this lemma implies that the phenomenon of automatic extension of unitary representations to trains is valid for infinite-dimensional $p$-adic groups., Comment: 15p

Having potential applications in multiplexing techniques and in the synthesis of frames, orthogonality (or strongly disjointness) plays a significant role in frame theory (e.g. construction of new frames from existing ones, constructions related with duality, etc.). In this article, we study orthogonality of a pair of frames over locally compact abelian (LCA) groups. We start with the investigation of the dual Gramian analysis tools of Ron and Shen through a pre-Gramian operator over the set-up of LCA groups. Then we fiberize some operators associated with Bessel families generated by unitary actions of co-compact (not necessarily discrete) subgroups of LCA groups. Using this fiberization, we study and characterize a pair of orthogonal frames generated by the action of a unitary representation ρ of a co-compact subgroup Γ ⊂ G on a separable Hilbert space L 2 ( G ) , where G is a second countable LCA group. Precisely, we consider frames of the form { ρ ( γ ) ψ : γ ∈ Γ , ψ ∈ Ψ } for a countable family Ψ in L 2 ( G ) . We pay special attention to this problem in the context of translation-invariant space by assuming ρ as the action of Γ on L 2 ( G ) by left-translation. The representation of Γ acting on L 2 ( G ) by (left-)translation is called the (left-)regular representation of Γ. Further, we apply our results on co-compact Gabor systems over LCA groups. At this juncture, it is pertinent to note that the resulting characterization can be useful for constructing new frames by using various techniques including the unitary extension principle by Ron and Shen [24] and its recent extension to LCA groups by Christensen and Goh [7] .

We introduce a Schr¨odinger model for the generic representations of a Heisenberg motion group and we construct adapted Weyl correspondences for these representations by adapting the method introduced in [ B. Cahen, Weyl quantization for semidirect products, Differential Geom. Appl. 25 (2007), 177-190].

We consider Banach spaces equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. Using these semigroups we introduce an analog of a modulus of continuity and define analogs of Besov norms. A generalization of a classical interpolation theorem is proven in which the role of Sobolev spaces is played by subspaces defined in terms of infinitesimal operators of these semigroups. We show that our assumptions about a given set of semigroups are satisfied in the case of a strongly continuous bounded representation of a Lie group. In the case of a unitary representation in a Hilbert space we consider an analog of the Laplace operator and use it to define Paley-Wiener vectors. It allows us to develop a generalization of the Shannon-type sampling in Paley-Wiener subspaces and to construct Paley-Wiener nearly Parseval frames in the entire Hilbert space. It is shown that Besov spaces defined previously in terms of the modulus of continuity can be described in terms of approximation by Paley-Wiener vectors and also in terms of the frame coefficients. Throughout the paper we extensively use theory of interpolation and approximation spaces. The paper ends with applications of our results to function spaces on homogeneous manifolds., Comment: arXiv admin note: text overlap with arXiv:1512.08668

Let $\Gamma_{w}$ be a non-cofinite Hecke triangle group with cusp width $w>2$ and let $\varrho\colon\Gamma_w\to U(V)$ be a finite-dimensional unitary representation of $\Gamma_w$. In this note we announce a new fractal upper bound for the Selberg zeta function of $\Gamma_{w}$ twisted by $\varrho$. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by $\exp\left( C_{\varepsilon} \vert s\vert^{\delta + \varepsilon} \right)$, where $\delta = \delta_{w}$ denotes the Hausdorff dimension of the limit set of $\Gamma_{w}$. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces $X=\widetilde{\Gamma}\backslash\mathbb{H}$ where $\widetilde{\Gamma}$ is a finite index, torsion-free subgroup of $\Gamma_w$., Comment: 10 pages, 1 figure

Let G be a connected, simply connected nilpotent group and π be an irreducible unitary representation of G that is square-integrable modulo its center Z ( G ) on L 2 ( R d ) . We prove that under reasonably weak conditions on G and π there exist a discrete subset Γ of G / Z ( G ) and some (relatively) compact set F ⊆ R d such that { | F | − 1 / 2 π ( γ ) 1 F | γ ∈ Γ } forms an orthonormal basis of L 2 ( R d ) . This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis. The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set Γ can be chosen to be a uniform subgroup of G / Z .

In this paper we study the characteristic polynomial of multiparameter pencil z 1 A 1 + z 2 A 2 + ⋯ + z s A s . The main theorem states that a unitary representation of a finitely generated group contains a one-dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor. It follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible. The result is of kin to the Dedekind and Frobenius theorem on finite group determinant.