On higher Frobenius-Schur indicators

Kashina, Yevgenia; Sommerhäuser, Yorck; Zhu, Yongchang

doi:10.1090/memo/0855

Cited by 105 publications

(181 citation statements)

References 41 publications

(103 reference statements)

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“…. , n} to be congruent to i + k modulo n. (This occurs for similar reasons in [10] as a special case of more general permutation constructions.) Note that s is an n-cycle since n and k are relatively prime.…”

Section: Frobenius-schur Endomorphismsmentioning

confidence: 85%

“…The higher indicators ν n (V ) of an irreducible group representation V , which have less obvious meaning for the structure of V , were generalized to simple modules of a semisimple Hopf algebra by Kashina, Sommerhäuser, and Zhu [10].…”

Section: Introductionmentioning

confidence: 99%

“…with the invariant subspace (V ⊗n ) H , and E (n) V is given by a cyclic permutation; the description of ν n (V ) as a trace in this case is contained in [10]. If n = 2 and V is simple, then C(I, V ⊗ V ) is one-dimensional or vanishes.…”

Section: Introductionmentioning

confidence: 99%

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Higher Frobenius-Schur indicators for pivotal categories

Ng¹,

Schauenburg²

2007

Contemporary Mathematics

128

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Abstract. We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a k-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a k-linear semisimple pivotal monoidal category -where both notions are defined -, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.

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Section: Frobenius-schur Endomorphismsmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

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Higher Frobenius-Schur indicators for pivotal categories

Ng¹,

Schauenburg²

2007

Contemporary Mathematics

128

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show abstract

“…The second holds by (206). The fourth holds by (186), since f ·u and u·f commute in the convolution algebra Hom Sp k (c · c, k · k) by (37). We complete the proof by justifying the last equality.…”

Section: In Additionmentioning

confidence: 71%

“…Let a i be a monoid and c i a comonoid, for i = 1, 2. If f i , g i : c i → a i are morphisms of species, then (37) (…”

Section: 7mentioning

confidence: 99%

Hopf monoids in the category of species

Aguiar¹,

Mahajan²

2013

Contemporary Mathematics

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Abstract. A Hopf monoid (in Joyal's category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a self-contained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and Dynkin idempotents and discuss the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems for Hopf monoids.

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Research and application of city ventilation assessments based on satellite data and GIS technology: a case study of the Yanqi Lake Eco‐city in Huairou District, Beijing

Liu

Fang

Cheng

et al. 2016

Meteorological Applications

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A method of performing city-scale ventilation assessments based on remote sensing (RS) and geographic information system (GIS) technology is proposed and applied effectively in an assessment of the local ventilation environment before and after the construction of the Yanqi Lake Eco-city in Huairou District, Beijing. The ventilation environment of a city is dependent on its background wind environment and ventilation potential. The background wind environment can be assessed through meteorological statistical analyses or numerical simulations, whereas the ventilation potential can be estimated using the underlying surface dynamic roughness length (RL) and the surrounding sky view factor (SVF). Based on the principle of morphology, the RL of a vegetation area is determined mainly by the vegetation types, vegetation canopy area index and vegetation height, whereas the RL of an urban area is defined mainly by the building area coverage and building height. The SVF can be estimated according to high-resolution digital raster elevation data. Based on the combination of the RL and SVF, the ventilation potential is graded, and the ventilation potential index (VPI) is defined and used to assess quantitatively the ventilation capacity of the city. The present study calculates the RL of the Yanqi Lake Eco-city using Landsat-TM5 satellite data from 26 July 2011, Geoscience Laser Altimeter System (GLAS) satellite data from 2005 and Beijing basic geographical data at 1:2000 from 2009. Subsequently, the RL and ventilation potential distribution of the Yanqi Lake Eco-city is mapped at a 100 m spatial resolution. The ventilation conditions before and after the construction of the Yanqi Lake Eco-city are evaluated using the VPI in combination with the background wind field. According to changes in the ventilation capacity, relevant policy suggestions for urban buildings and design factors for ventilated corridors are postulated.

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On higher Frobenius-Schur indicators

Cited by 105 publications

References 41 publications

Higher Frobenius-Schur indicators for pivotal categories

Higher Frobenius-Schur indicators for pivotal categories

Hopf monoids in the category of species

Research and application of city ventilation assessments based on satellite data and GIS technology: a case study of the Yanqi Lake Eco‐city in Huairou District, Beijing

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