Homotopy equivalent group representations and Picard groups op the burnside ring and the character ring

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

Glasgow Mathematical Journal ◽

10.1017/s0017089521000215 ◽

2021 ◽

pp. 1-8

Author(s):

DANIEL KASPROWSKI ◽

MARKUS LAND

Keyword(s):

Fundamental Group ◽

Homotopy Equivalent ◽

Poincaré Duality ◽

Poincare Duality ◽

Canonical Map ◽

Duality Space ◽

Simply Connected Manifolds ◽

Simply Connected ◽

Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

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Crystallography: Symmetry groups and group representations

EPJ Web of Conferences ◽

10.1051/epjconf/20122200006 ◽

2012 ◽

Vol 22 ◽

pp. 00006

Author(s):

B. Grenier ◽

R. Ballou

Keyword(s):

Symmetry Groups ◽

Group Representations

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Probing charge pumping and relaxation of the chiral anomaly in a Dirac semimetal

Science Advances ◽

10.1126/sciadv.abg0914 ◽

2021 ◽

Vol 7 (16) ◽

pp. eabg0914

Author(s):

Bing Cheng ◽

Timo Schumann ◽

Susanne Stemmer ◽

N. P. Armitage

Keyword(s):

Terahertz Spectroscopy ◽

Point Group ◽

Spectral Weight ◽

Chiral Anomaly ◽

Group Representations ◽

Field Independence ◽

Transport Channel ◽

Scattering Rate ◽

Chiral Magnetic Effect ◽

A Charge

The linear band crossings of 3D Dirac and Weyl semimetals are characterized by a charge chirality, the parallel or antiparallel locking of electron spin to its momentum. These materials are believed to exhibit an E · B chiral magnetic effect that is associated with the near conservation of chiral charge. Here, we use magneto-terahertz spectroscopy to study epitaxial Cd3As2 films and extract their conductivities σ(ω) as a function of E · B. As field is applied, we observe a markedly sharp Drude response that rises out of the broader background. Its appearance is a definitive signature of a new transport channel and consistent with the chiral response, with its spectral weight a measure of the net chiral charge and width a measure of the scattering rate between chiral species. The field independence of the chiral relaxation establishes that it is set by the approximate conservation of the isospin that labels the crystalline point-group representations.

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