Augmentations and link group representations

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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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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Group representations which vanish at infinity

Mathematische Annalen ◽

10.1007/bf01536185 ◽

1980 ◽

Vol 251 (2) ◽

pp. 185-190 ◽

Cited By ~ 3

Author(s):

Keith F. Taylor

Keyword(s):

Group Representations

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Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500438 ◽

2018 ◽

Vol 27 (06) ◽

pp. 1850043 ◽

Cited By ~ 1

Author(s):

Paul P. Gustafson

Keyword(s):

Braid Group ◽

Mapping Class Group ◽

Class Group ◽

Braid Groups ◽

Fusion Category ◽

Mapping Class ◽

Group Representations ◽

Colored Graphs ◽

Finite Image ◽

Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

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Interpersonal Disgust, Ideological Orientations, and Dehumanization as Predictors of Intergroup Attitudes

Psychological Science ◽

10.1111/j.1467-9280.2007.01962.x ◽

2007 ◽

Vol 18 (8) ◽

pp. 691-698 ◽

Cited By ~ 207

Author(s):

Gordon Hodson ◽

Kimberly Costello

Keyword(s):

Social Dominance ◽

Social Dominance Orientation ◽

Path Model ◽

Intergroup Attitudes ◽

Right Wing ◽

Disgust Sensitivity ◽

Group Representations ◽

Negative Attitudes ◽

Attitudes Toward Immigrants ◽

Right Wing Authoritarianism

Disgust is a basic emotion characterized by revulsion and rejection, yet it is relatively unexamined in the literature on prejudice. In the present investigation, interpersonal-disgust sensitivity (e.g., not wanting to wear clean used clothes or to sit on a warm seat vacated by a stranger) in particular predicted negative attitudes toward immigrants, foreigners, and socially deviant groups, even after controlling for concerns with contracting disease. The mechanisms underlying the link between interpersonal disgust and attitudes toward immigrants were explored using a path model. As predicted, the effect of interpersonal-disgust sensitivity on group attitudes was indirect, mediated by ideological orientations (social dominance orientation, right-wing authoritarianism) and dehumanizing perceptions of the out-group. The effects of social dominance orientation on group attitudes were both direct and indirect, via dehumanization. These results establish a link between disgust sensitivity and prejudice that is not accounted for by fear of infection, but rather is mediated by ideological orientations and dehumanizing group representations. Implications for understanding and reducing prejudice are discussed.

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