Browsing by Subject "Biset functor"

Now showing 1 - 4 of 4

Open Access
Alcahestic subalgebras of the alchemic algebra and a correspondence of simple modules
(Elsevier Inc., 2008) Coşkun, Olcay
The unified treatment of the five module-theoretic notions, transfer, inflation, transport of structure by an isomorphism, deflation and restriction, is given by the theory of biset functors, introduced by Bouc. In this paper, we construct the algebra realizing biset functors as representations. The algebra has a presentation similar to the well-known Mackey algebra. We adopt some natural constructions from the theory of Mackey functors and give two new constructions of simple biset functors. We also obtain a criterion for semisimplicity in terms of the biset functor version of the mark homomorphism. The criterion has an elementary generalization to arbitrary finite-dimensional algebras over a field.
Open Access
Deformations of some biset-theoretic categories
(2020-09) Öğüt, İsmail Alperen
We define the subgroup category, a category on the class of finite groups where the morphisms are given by the subgroups of the direct products and the composition is the star product. We also introduce some of its deformations and provide a criteria for their semisimplicity. We show that biset category can be realized as an invariant subcategory of the subgroup category, where the composition is much simpler. With this correspondence, we obtain some of the deformations of the biset category. We further our methods to the fibred biset category by introducing the subcharacter partial category. Similarly, we also realize the fibred biset category and some of its deformations in a category where the composition is more easily described.
Open Access
A filtration of the modular representation functor
(Academic Press, 2007) Yaraneri, E.
Let F and K be algebraically closed fields of characteristics p > 0 and 0, respectively. For any finite group G we denote by K RF (G) = K ⊗Z G0 (F G) the modular representation algebra of G over K where G0 (F G) is the Grothendieck group of finitely generated F G-modules with respect to exact sequences. The usual operations induction, inflation, restriction, and transport of structure with a group isomorphism between the finitely generated modules of group algebras over F induce maps between modular representation algebras making K RF an inflation functor. We show that the composition factors of K RF are precisely the simple inflation functors SC, Vi where C ranges over all nonisomorphic cyclic p′-groups and V ranges over all nonisomorphic simple K Out (C)-modules. Moreover each composition factor has multiplicity 1. We also give a filtration of K RF. © 2007 Elsevier Inc. All rights reserved.
Open Access
Rhetorical biset functors, rational p-biset functors and their semisimplicity in characteristic zero
(Academic Press, 2008) Barker, Laurence
Rhetorical biset functors can be defined for any family of finite groups that is closed under subquotients up to isomorphism. The rhetorical p-biset functors almost coincide with the rational p-biset functors. We show that, over a field with characteristic zero, the rhetorical biset functors are semisimple and, furthermore, they admit a character theory involving primitive characters of automorphism groups of cyclic groups.