Browsing by Subject "Biset category"

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    Blocks of Mackey categories
    (Elsevier, 2016) Barker, L.
    For a suitable small category F of homomorphisms between finite groups, we introduce two subcategories of the biset category, namely, the deflation Mackey category MF← and the inflation Mackey category MF→. Let G be the subcategory of F consisting of the injective homomorphisms. We shall show that, for a field K of characteristic zero, the K-linear category KMG=KMG←=KMG→ has a semisimplicity property and, in particular, every block of KMG owns a unique simple functor up to isomorphism. On the other hand, we shall show that, when F is equivalent to the category of finite groups, the K-linear categories KMF← and KMF→ each have a unique block. © 2015 Elsevier Inc.
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    Simple functors of admissible linear categories
    (Springer, 2016) Barker, L.; Demirel, M.
    Generalizing an idea used by Bouc, Thévenaz, Webb and others, we introduce the notion of an admissible R-linear category for a commutative unital ring R. Given an R-linear category (Formula presented.) , we define an (Formula presented.) -functor to be a functor from (Formula presented.) to the category of R-modules. In the case where (Formula presented.) is admissible, we establish a bijective correspondence between the isomorphism classes of simple functors and the equivalence classes of pairs (G, V) where G is an object and V is a module of a certain quotient of the endomorphism algebra of G. Here, two pairs (F, U) and (G, V) are equivalent provided there exists an isomorphism F ← G effecting transport to U from V. We apply this to the category of finite abelian p-groups and to a class of subcategories of the biset category. © 2015, Springer Science+Business Media Dordrecht.