Trace maps for Mackey algebras

JOURNAL OF ALGEBRA

卷:426

Article

Rognerud, Baptiste

关键词: Finite group;    Mackey functor;    Symmetric algebra;    Symmetric monoidal category;    Burnside ring;   

DOI  :  10.1016/j.jalgebra.2014.12.021

来源: Elsevier

【 摘 要 】

Let G be a finite group and R be a commutative ring. The Mackey algebra mu(R)(G) shares a lot of properties with the group algebra RG however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetry of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra. Using the fact that the category of Mackey functors is a closed symmetric monoidal category, we prove that the Mackey algebra mu(R)(G) is a symmetric algebra if and only if the family of Burnside algebras (RB(H))(H <= G) is a family of symmetric algebras with a compatibility condition. As a corollary, we recover the well known fact that over a field of characteristic zero, the Mackey algebra is always symmetric. Over the ring of integers the Mackey algebra of G is syrnmetric if and only if the order of G is square free. Finally, if (K, O, k) is a p-modular system for G, we show that the Mackey algebras mu(O)(G) and mu(k)(G) are symmetric if and only if the Sylow p-subgroups of G are of order 1 or p. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free