Introduction to the Representation Theory of Compact and Locally Compact Groups

Robert, Alain

doi:10.1017/cbo9780511661891

Cited by 47 publications

(27 citation statements)

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“…the spectral projections as in [11,15.12]) we get the next proposition, which yields another proof of Mackey's criteria for irreducibility and disjointness. Proposition 1.3.…”

Section: Elementary Intertwining Operatorssupporting

confidence: 53%

On Induced Representations of Discrete Groups

Binder¹

1993

Proceedings of the American Mathematical Society

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Abstract.The paper deals with induced representations ind^ cr of a locally compact group G where H is an open subgroup. Using "elementary intertwining operators", we first describe the commutant ind# o(G)' (also in the case of realizing the induced representations with positive definite measures). Then criteria for irreducibility and pairwise disjointness of induced representations are given. Finally, special attention is devoted to abelian subgroups H .

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“…the spectral projections as in [11,15.12]) we get the next proposition, which yields another proof of Mackey's criteria for irreducibility and disjointness. Proposition 1.3.…”

Section: Elementary Intertwining Operatorssupporting

confidence: 53%

On Induced Representations of Discrete Groups

Binder¹

1993

Proceedings of the American Mathematical Society

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“…[47,Sec. 19]): We interprete Γ → A\Γ as covering space with abelian covering group A acting on Γ from the left; the corresponding translation action T a on 2 (Γ) coincides with the left regular representation L a (a ∈ A).…”

Section: Non-abelian Floquet Transformationmentioning

confidence: 99%

Existence of Spectral Gaps, Covering Manifolds and Residually Finite Groups

Lledó

Post

2008

Rev. Math. Phys.

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In the present paper we consider Riemannian coverings (X, g) → (M, g) with residually finite covering group Γ and compact base space (M, g). In particular, we give two general procedures resulting in a family of deformed coverings (X, gε) → (M, gε) such that the spectrum of the Laplacian ∆ (Xε,gε) has at least a prescribed finite number of spectral gaps provided ε is small enough.If Γ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that spec ∆ (X,gε) has, in addition, band-structure and there is an asymptotic estimate for the number N (λ) of components of spec ∆ (X,gε) that intersect the interval [0, λ]. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.

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“…The concept of group convolution of functions on a wide variety of abstract groups is well known in pure mathematics literature [40]. Although many of the engineering applications involving group convolutions were developed independently, they can be systematically formulated within a group-invariant system-theoretic framework [21], [41], [64], [65].…”

Section: The Concept Of Convolution and Fourier Analysis On Groupsmentioning

confidence: 99%

“…As a result, the following inversion formula holds [40]: (3.9a) or in component form (3.9b) where denotes the dimension of the representation . The Plancherel formula is given by (3.9c) For the case of separable locally compact groups of Type I, the Fourier inversion and the Plancherel formulas are given by [49] Then, the Fourier transform of a function defined on the affine group, i.e., is given by the following operator valued functions:…”

Section: B Fourier Analysis On Groupsmentioning

confidence: 99%

Stochastic Deconvolution Over Groups

Yazıcı

2004

IEEE Trans. Inform. Theory

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Abstract-In this paper, we address a class of inverse problems that are formulated as group convolutions. This is a rich area of research with applications to Radon transform inversion for tomography, wide-band and narrow-band signal processing, inverse rendering in computer graphics, and channel estimation in communications, as well as robotics and polymer science. We present a group-theoretic framework for signal modeling and analysis for such problems and propose a minimum mean-square error (MMSE) deconvolution method in a probabilistic setting. Key components of our approach are group representation theory and the concept of group stationarity. The roposed deconvolution method incorporates a priori information and noise statistics into the inversion process, which leads to a natural regularized solution. We present recovery of self-similar processes that are "blurred" and embedded in noise as a demonstration example. The method is applicable to a wide range of inverse problems involving both commutative and noncommutative groups including finite, compact, and majority of well-behaved locally compact groups.

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Introduction to the Representation Theory of Compact and Locally Compact Groups

Cited by 47 publications

References 0 publications

On Induced Representations of Discrete Groups

On Induced Representations of Discrete Groups

Existence of Spectral Gaps, Covering Manifolds and Residually Finite Groups

Stochastic Deconvolution Over Groups

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