Inductions, restrictions, evaluations and subfunctors of mackey functors
Mackey funktorların genişletilmesi, kısıtlanması, hesaplanması, ve alt funktorları
- Tez No: 177182
- Danışmanlar: DOÇ. DR. LAURENCE J. BARKER
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Mackey functor, Mackey algebra, simple, indecomposable, restriction, induction, evaluation, socle, radical, Clifford?s theorem, Green?s indecomposibilitytheorem, maximal subfunctor, Brauer quotient, minimal subfunctor, restrictionkernel, primordial, coprimordial, uniserial, Burnside functor, socle series, radical series, composition series, composition factors, Loewy lenght, fixed pointfunctor, functors for p-groups
- Yıl: 2008
- Dil: İngilizce
- Üniversite: İhsan Doğramacı Bilkent Üniversitesi
- Enstitü: Mühendislik ve Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Matematik Bölümü
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Özet yok.
Özet (Çeviri)
In this thesis we try to relate the subfunctor structure of a given Mackey functorM for a finite group G to the submodule structure of the KNG(H)-module M(H)where H is a subgroup of G.We mainly study the socle and the radical of a Mackey functor M for a finitegroup G over a field K, (usually, of characteristic p > 0). For a subgroup Hof G, we construct bijections between some classes of the simple subfunctors ofM and some classes of the simple KNG(H)-submodules of M(H). We relate themultiplicity of a simple Mackey functor SGH,V in the socle of M to the multiplicityof V in the socle of a certain KNG(H)-submodule of M(H). We also obtainsimilar results for the maximal subfunctors of M. We specialize our results tosome specific kinds of Mackey functors for G that includes the functors obtainedby inducing or restricting a simple Mackey functor, Mackey functors for a p-group,the fixed point functor, and the Burnside functor BGK.Let M be the Mackey functor“GK SKH,W for G obtained by inducing a simpleMackey functor SKH,W for K. For example, we observe that the socle and theradical of M can be determined from the socle and the radical of the KNG(H)-module V =”NG(H)NK(H)W. We also find similar results for Mackey functors obtainedby restricting a simple Mackey functor. Moreover, we derive criterions for aMackey functor obtained by inducing or restricting a simple Mackey functor tobe simple, semisimple or indecomposable.Our results about induced or restricted Mackey functors include Mackey functorversions of two classical and frequently used results in the representationtheory finite groups, namely Clifford?s theorem and Green?s indecomposibilitytheorem.We also apply our general results to Mackey functors satisfying some specialconditions such as having a unique maximal or simple subfunctor, being uniserial,and being a functor for a p-group. We give some results about primordial andcoprimordial subgroups of G for such kind of functors, and we refine our generalresults and obtain, for instance, a criterion for a Mackey functor to be a quotientof a projective Mackey functor, and find some information about compositionseries.In later chapters the main Mackey functor to which we apply our generalresults is the Burnside functor BGK. We first find the maximal subfunctors of BGKfor any group G, and obtain some results about evaluations of the terms of theradical series of BGK. We also get some results about simple Mackey functors inradical layers of BGKwhose minimal subgroups are p-subgroup of G. Assumingthat G is a p-group we find the first four top factors of the radical series of BGK,and assuming further that G is an abelian p-group we find the radical series ofBGKcompletely, which means that in this case we find the evaluations of the termsof the radical series, and the simple Mackey functors appearing in radical layers,and the Loewy length of BGK. We also study the socle series of BGK. This seems tobe harder than the radical series. Nevertheless, we obtain similar results for thesocle series of BGKassuming mostly that G is an abelian p-group. To illustrateapplications of our general results we also study briefly the radical and the socleseries of fixed point functors FPGV where V is a one dimensional KG-module.We finish this thesis by trying to find possible relations between the socles andthe radicals of the Mackey functors of the form T and FT where T is a Mackeyfunctor and F is one of the functors restriction, inflation, evaluation, or adjointsof them, between Mackey functor categories.
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