scholarly journals Support varieties for rational representations

2014 ◽  
Vol 151 (4) ◽  
pp. 765-792 ◽  
Author(s):  
Eric M. Friedlander
Keyword(s):  
Finite Groups ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Infinite Dimensional ◽  
Group Schemes ◽  
The Family ◽  
Close Relationship ◽  
Frobenius Kernels ◽  
Support Varieties ◽  
Finite Group Schemes

We introduce support varieties for rational representations of a linear algebraic group $G$ of exponential type over an algebraically closed field $k$ of characteristic $p>0$. These varieties are closed subspaces of the space $V(G)$ of all 1-parameter subgroups of $G$. The functor $M\mapsto V(G)_{M}$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between $V(G)_{M}$ and the family of support varieties $V_{r}(G)_{M}$ obtained by restricting the $G$ action to Frobenius kernels $G_{(r)}\subset G$. These support varieties seem particularly appropriate for the investigation of infinite-dimensional rational $G$-modules.

scholarly journals Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes

2018 ◽  
Vol 30 (2) ◽  
pp. 479-495
Author(s):  
Yang Pan
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Finite Groups ◽  
Positive Characteristic ◽  
Group Scheme ◽  
Closed Field ◽  
Group Schemes ◽  
Frobenius Kernel ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractWe investigate the saturation rank of a finite group scheme defined over an algebraically closed field{\Bbbk}of positive characteristicp. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group{\operatorname{SL}_{n}}.

Finite group schemes of p-rank ≤ 1

2017 ◽  
Vol 166 (2) ◽  
pp. 297-323
Author(s):  
HAO CHANG ◽  
ROLF FARNSTEINER
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Group Scheme ◽  
Closed Field ◽  
Difficult Case ◽  
Modular Representation Theory ◽  
Group Schemes ◽  
Restricted Lie Algebras ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractLet be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

scholarly journals Restrictions to G(𝔽p) and G(r) of rational G-modules

2011 ◽  
Vol 147 (6) ◽  
pp. 1955-1978 ◽  
Author(s):  
Eric M. Friedlander
Keyword(s):  
Algebraic Group ◽  
Perfect Field ◽  
Reductive Algebraic Group ◽  
Finite Dimensional ◽  
The Family ◽  
Frobenius Kernels ◽  
Support Varieties ◽  
The Relationship

AbstractWe fix a prime p and consider a connected reductive algebraic group G over a perfect field k which is defined over 𝔽p. Let M be a finite-dimensional rational G-module M, a comodule for k[G]. We seek to somewhat unravel the relationship between the restriction of M to the finite Chevalley subgroup G(𝔽p)⊂G and the family of restrictions of M to Frobenius kernels G(r) ⊂G. In particular, we confront the conundrum that if M is the Frobenius twist of a rational G-module N,M=N(1), then the restrictions of M and N to G(𝔽p) are equal whereas the restriction of M to G(1) is trivial. Our analysis enables us to compare support varieties (and the finer non-maximal support varieties) for G(𝔽p) and G(r) of a rational G-module M where the choice of r depends explicitly on M.

Resolving the Lourinia armata (Claus, 1866) complex with remarks on the monophyletic status of Louriniidae, Monard 1927 (Copepoda: Harpacticoida)

Zootaxa ◽  
2021 ◽  
Vol 5051 (1) ◽  
pp. 346-386
Author(s):  
SÜPHAN KARAYTUĞ ◽  
SERDAR SAK ◽  
ALP ALPER ◽  
SERDAR SÖNMEZ
Keyword(s):  
Comparative Evaluation ◽  
Species Complex ◽  
Detailed Examination ◽  
The Other ◽  
Phylogenetic Position ◽  
Nicobar Island ◽  
Other Hand ◽  
The Family ◽  
Close Relationship ◽  
Almost All

An attempt was made to test if Lourinia armata (Claus, 1866)—as it is currently diagnosed—represents a species complex. Detailed examination and comparisons of several specimens collected from different localities suggest that L. armata indeed represents a complex of four closely related morphospecies that can be differentiated from one another by only detailed observations. One of the four species is identified as Lourinia aff. armata and the other three species are described as new to science and named as Lourinia wellsi sp. nov., L. gocmeni sp. nov., and L. aldabraensis sp. nov. Detailed review of previous species records indicates that the genus Lourinia Wilson, 1924 is distributed worldwide. Ceyloniella nicobarica Sewell, 1940, originally described from Nicobar Island and previously considered a junior subjective synonym of L. armata is reinstated as Lourinia nicobarica (Sewell, 1940) comb. nov. on the basis of the unique paddle-shaped caudal ramus seta V. It is postulated that almost all of these records are unreliable in terms of representing true Lourinia aff. armata described herein. On the other hand, the comparative evaluation of the illustrations and descriptions in the published literature indicates the presence of several new species waiting to be discovered in the genus Lourinia.                 It has been determined that, according to updated modern keys, the recent inclusion of the monotypic genus Archeolourinia Corgosinho & Schizas, 2013 in the Louriniidae is not justified since Archeolourinia shermani Corgosinho & Schizas, 2013 does not belong to this family but should be assigned to the Canthocamptidae. On the other hand, it has been argued that the exact phylogenetic position of the Louriniidae still remains problematic since none of the diagnostic characters supports the monophyly of the family within the Oligoarthra. It has also been argued that the close relationship between Louriniidae and Canthocamptidae is supported since both families share the homologous sexual dimorphism (apophysis) on P3 endopod. The most important characteristic that can possibly be used to define Louriniidae is the reduction of maxilliped.  

Morphology of a unique ophidiid, Hypopleuron caninum Radcliffe 1913 (Ophidiiformes, Ophidiidae, Neobythitinae), suggesting a close relationship with the family Carapidae

Zootaxa ◽  
2018 ◽  
Vol 4521 (4) ◽  
pp. 499
Author(s):  
SHINPEI OHASHI
Keyword(s):  
Phylogenetic Position ◽  
Dorsal Fin ◽  
The Family ◽  
Close Relationship ◽  
Pelvic Fin

Osteological and myological characters of the ophidiid Hypopleuron caninum are described here in detail. In addition to well-known characters of Ophidiidae (e.g., anal-fin origin well posterior to dorsal-fin origin; barbel-like pelvic fin; many scales), many unique or rare conditions were also recognized, including predorsal bone and maxilla condition, and canine teeth on the premaxilla, which were suggestive of a relationship with Carapidae. Furthermore, the predorsal bone appear to be homologous with the vexillar support, a carapid synapomorphy. Although several characters differ from carapid counterparts (e.g., more anterior anal-fin origin and scales absent in Carapidae), some intermediate conditions between Ophidiidae and Carapidae are apparent in Hypopleuron caninum, which may therefore occupy an intermediate phylogenetic position between the two families. A unique morphology of small second infraorbital bone is recognized in all ophidiiform species observed in this study, and it may be a synapomorphic character of the order Ophidiiformes. 

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.

scholarly journals Stratification for module categories of finite group schemes

2017 ◽  
Vol 31 (1) ◽  
pp. 265-302 ◽  
Author(s):  
Dave Benson ◽  
Srikanth B. Iyengar ◽  
Henning Krause ◽  
Julia Pevtsova
Keyword(s):  
Finite Group ◽  
Group Schemes ◽  
Finite Group Schemes

Comparative study of experienced and anticipated stigma in Serbia and the world

2011 ◽  
Vol 58 (4) ◽  
pp. 355-361 ◽  
Author(s):  
Ivona Milačić Vidojević ◽  
Dragana Ðurić Jočić ◽  
Oliver Tošković
Keyword(s):  
International Study ◽  
Demographic Variables ◽  
Personal Safety ◽  
Main Study ◽  
Individual Interviews ◽  
The Family ◽  
Close Relationship ◽  
Positive Discrimination ◽  
Persons With Schizophrenia ◽  
Anticipated Stigma

Aim: The presented study is part of the International Study of Discrimination and Stigma Outcomes (INDIGO) conducted in 28 countries. The aim of this study was to compare the degree of experienced and anticipated stigma of patients with schizophrenia in Serbia with world results. Method: The Discrimination and Stigma Scale (DISC) was applied in individual interviews. The sample comprised 732 subjects of the main study and 50 subjects from Serbia. Results: The amount of negative experienced discrimination in Serbia is the same as in other participating countries. The Serbian sample shows more negative discrimination in intimate relationships, personal safety and general harm due to the diagnosis, but more positive discrimination within the family, privileges during treatment of somatic illnesses, and benefits enjoyed in social and retirement insurance. The Serbian sample shows less anticipated discrimination in looking for a close relationship and in stopping self from applying for work. The results show that socio-demographic variables could not predict negative experienced and anticipated discrimination, while those same variables enable the prediction of 34% of positive discrimination. Conclusion: Compared to other investigated countries, stigmatization processes for persons with schizophrenia do not differ considerably in Serbia, which justifies the implementation of unique anti-stigma programmes and joint upgrading of legal regulations.

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