DefiningkinG(k)

AbstractWe show how the field of definitionkof ak-isotropic absolutely almost simplek-groupG“lives” in the groupG(k) ofk-rational points. The construction which is inspired by the fundamental work of Borel-Tits is as follows: We choose an element inside the center of the unipotent radical of a minimal parabolick-subgroupP; the orbit under the action of the centerZof a Levik-subgroup ofPgenerates a one-dimensional vector space which then carries the additive field structure in a natural way. The multiplicative structure is induced by the action ofZ. IfGisk-simple, our construction yields a finite extensionlofk.As an immediate consequence we obtain an answer to a question of Borovik–Nesin under the additional assumption thatGisk-isotropic:Theorem. IfGis ak-simplek-isotropic group such thatG(k) has finite Morley rank, thenkis either algebraically closed or real closed. IfGis absolutely simplek-isotropic, thenkis algebraically closed

The orbit structure of Dynkin curves

Let G be a simple algebraic group over an algebraically closed field k; assume that Char k is zero or good for G. Let \cB be the variety of Borel subgroups of G and let e in Lie G be nilpotent. There is a natural action of the centralizer C_G(e) of e in G on the Springer fibre \cB_e = {B' in \cB | e in Lie B'} associated to e. In this paper we consider the case, where e lies in the subregular nilpotent orbit; in this case \cB_e is a Dynkin curve. We give a complete description of the C_G(e)-orbits in \cB_e. In particular, we classify the irreducible components of \cB_e on which C_G(e) acts with finitely many orbits. In an application we obtain a classification of all subregular orbital varieties admitting a finite number of B-orbits for B a fixed Borel subgroup of G.Comment: 12 pages, to appear in Math

Canonical discriminant analysis of the fatty acid profile of muscle to authenticate beef from grass-fed and other beef production systems: Model development and validation

peer-reviewedThe potential of diet-induced differences in the fatty acid profile of muscle to discriminate beef from different feeding systems and its potential use as an authentication tool was investigated. Three canonical discriminant models were built and validated using the fatty acid profile of beef from animals fed solely on pasture or cereal-based concentrates for 11 months or on various pasture/grass silage/concentrate combinations, including concentrates enriched with plant oils. Results indicated that models could successfully discriminate between grass-, partially grass- and concentrate-fed beef (accuracy = 99%) and between grass-fed beef and beef from animals supplemented with plant oils (accuracy = 96%). The approach also showed potential for distinguishing between beef from exclusively pasture-fed cattle and beef from cattle fed on pasture preceded by a period on ensiled grass (accuracy = 89%). Models were also applied to beef samples from 9 different countries. Of 97 international samples, including samples stated to be grass-fed, only 5% were incorrectly classified as Irish-grass-fed beef. These results suggested that the models captured traits in the fatty acid profile that are characteristic of Irish grass-fed beef and that this feature could be used for distinguishing Irish grass-fed beef from beef from other regions

On a question of Külshammer for representations of finite groups in reductive groups

Acknowledgments. The authors acknowledge the financial support of EPSRC Grant EP/L005328/1, Marsden Grants UOC1009 and UOA1021, and the DFG-priority programme SPP1388 “Representation Theory”. We are grateful to the referee for helpful suggestions.Peer reviewedPostprin

On Parabolic Subgroups Of Classical Groups With A Finite Number Of Orbits On The Unipotent Radical

Let G be a classical algebraic group defined over an algebraically closed field of characteristic zero. In case the simple factors of G are of type An , Bn or Cn , all parabolic subgroups P of G with a finite number of orbits on the unipotent radical Pu are determined: this is the case precisely when the nilpotent class of Pu is at most four. For groups of type Dn we obtain partial results depending on the structure of P . In this first of two papers we present the main theorem as well as a reduction of the problem for arbitrary simple classical groups to those of type An . The proofs for the cases of nilpotent class three and four in type An will appear in a subsequent paper. This is achieved by translating the group theoretic problem into a matrix problem within a Krull-Schmidt category where it is solved. There we also provide an explicit algorithm for enumerating the orbits in terms of certain diagrams

On Maximal Subgroups Of Finite Classical Groups

In [1] M. Aschbacher described the structure of the possible maximal subgroups of a classical group G with natural module M . Each such subgroup is either a member of a canonical class of subgroups, or it is the normalizer of a quasi-simple, absolutely irreducible subgroup H of G. In the latter case suppose that H is a classical group whose defining characteristic is coprime to that of G. The aim of this article is to help classify those intermediate groups H K G, where K is again a classical group whose defining characteristic is equal to the one of G. If H is linear, then under mild hypotheses on the rank of H and the size of the field Fq of definition of H we show that there are no such proper K