Tarix: 23.01.2025
Saat: 10.00
Məkan: Rəqəmsal Tədqiqat Laboratoriyası, Bakı Dövlət Universiteti, Əsas Bina, 3-cü mərtəbə
Diagonal $p$-permutation functors and local-global conjectures in block theory
Deniz Yilmaz, Bilkent University
Xülasə:
The local-global principle in modular representation theory asserts that the invariants of blocks of finite groups are determined by the invariants (of blocks) of local subgroups. There are several outstanding conjectures revolving around this principle. Alperin's block-wise weight conjecture (1987) predicts that the number of simple modules of a block is equal to the number of its weights. The finiteness conjectures of Donovan (1980) and Puig (1982) state that there are only finitely many blocks of finite groups with a given defect group, up to Morita and splendid Morita equivalence, respectively.
In this talk, we prove a finiteness theorem in the spirit of Donovan's and Puig's conjectures, in terms of functorial equivalences. We also give a reformulation of Alperin's conjecture in terms of diagonal $p$-permutation functors. Some parts of this work are joint with Boltje and Bouc.
Tarix: 05.12.2024
Saat: 10.30
Məkan: Rəqəmsal Tədqiqat Laboratoriyası, Bakı Dövlət Universiteti, Əsas Bina, 3-cü mərtəbə
Burnside Halqları Bölməsinin Green Biset funktorları üzərindəki sadə funktorları I
Ruslan Müslümov, ADA Universiteti
Tarix: 27.11.2024
Saat: 11.00
Məkan: Rəqəmsal Tədqiqat Laboratoriyası, Bakı Dövlət Universiteti, Əsas Bina, 3-cü mərtəbə
Burnside Halqları Bölməsinin Green Biset funktorları üzərindəki sadə funktorları II
Ruslan Müslümov, ADA Universiteti
Xülasə:
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.
Tarix: 24.10.2024
Saat: 10.00
Məkan: Rəqəmsal Tədqiqat Laboratoriyası, Bakı Dövlət Universiteti, Əsas Bina, 3-cü mərtəbə
Biset Funktorlarının Yapışdırılması üçün Maneələr I
Olcay Coşkun, MRL
Xülasə:
This is the first of a series of seminars in which we develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a biset functor defined on the subquotients of a finite group \(G\). The obstruction groups for this theory are the reduced cohomology groups of a category $D^*_G$ whose objects are the sections $(U, V)$ of $G$, where $V$ is a non-trivial normal subgorup of the subgroup $U$ of $G$, and whose morphisms are defined as a generalization of morphisms in the orbit category.
In this talk, we shall start introducing the basic setup for the upcoming talks.