Simple functors over the Green biset functor of section Burnside rings

by Ruslan Muslumov, Ada University

Abstract:

Biset functors over a commutative and unitary ring $k$ provide a powerful framework for studying finite groups and their actions. The biset category, whose objects are finite groups and morphism sets are given by Grothendieck groups $B(G, H)$ of finite $(G, H)$-bisets, serves as the foundation for this theory. The study of biset functors has yielded remarkable results, such as the evaluation of the Dade group of endo-permutation modules of a $p$-group and the determination of the unit group of the Burnside ring of a $p$-group.

To delve deeper into the intricate structure of biset functors, one can explore ring objects within this category. This pursuit leads to a more sophisticated algebraic structure known as a Green biset functor. This extension enables a richer understanding of the interplay between algebraic structures and group actions, offering a broader perspective on the relationships between finite groups and their associated algebraic objects.

Serge Bouc made significant contributions to this field by introducing the slice Burnside ring and the section Burnside ring for a finite group $G$. These rings encapsulate essential information about the group and its actions, providing a nuanced algebraic viewpoint. Bouc’s work demonstrated that both the slice Burnside ring and the section Burnside ring naturally possess the structure of a Green biset functor, showcasing the versatility of biset functor theory in capturing and characterizing algebraic structures arising from group actions.

The classification of simple modules over the section Burnside ring of $G$ represents a further advancement in this line of research. This classification is achieved through the application of the fibered biset functor approach, as detailed in the article by Robert Boltje and Olcay Coskun. This approach provides a systematic and powerful method for understanding the intricate module structure associated with the section Burnside ring. By leveraging the tools and concepts from biset functor theory, the classification sheds light on the representation theory of finite groups and offers valuable insights into the algebraic structure underlying group actions.

In summary, the study of biset functors and their algebraic counterparts, such as Green biset functors, slice Burnside rings, and section Burnside rings, provides a rich and fruitful framework for understanding the algebraic structures arising from group actions. The classification of simple modules over the section Burnside ring, achieved through the fibered biset functor approach, represents a significant contribution to this area of research, advancing our knowledge of the intricate relationships between algebra and group theory.

This is a joint work with Olcay Coskun.

Date: November 27, 2024 
Time: 11.00 
Location: Digital Research Lab, Baku State University, Main Building, 3rd floor