Abstract
Let G be a finite group and p a prime. The Brown complex of G at the prime p is the simplicial complex associated with the poset of non-trivial p-subgroups of G. The topology of this complex plays a crucial role in understanding the algebraic p-local structure of the group and has significant implications for related representation theoretic questions. Quillen has shown that the Brown complex of a finite reductive group defined over a field of characteristic p is homotopy equivalent to the Tits building. In this talk, we consider the case of a finite reductive group defined over a field of characteristic different from p and show that the homotopy type of the Brown complex can be described in terms of the generic Sylow theory introduced by Broué and Malle.
Talk
We give a combinatorial defintion of Simplizialkomplex. An abstract example is the collection of totally ordered sets in a poset. To a simplicial complex we consider the geometric realization.
The brown complex
Let be a finite groups, prime number, poset of -subgroups. The brown complex contains information on how the subgroups are related.
There is a roposition which allows us to find a Strongly p-embedded subgroup if the complex is disconncted.
A proof by quillen gives a requirement for when a groups is contractible.
AIM: We wanto to describe a Finite group of Lie type Let prime power, connected Reductive group defined over . A finite groups of Lie type is the set of rational points (I dont get it)
There is a connection to the Bruhat-Tits building.
We define a polynomial form Cyclotomic polynomials. Some extra steps then give us a -split Levi subgroup. From that we get a new simplicial complex.
A theorem by Rossi 2023 tells us that the simplicial complexes of to groups objects are Homotopy equivalent.
We present the Alperin Weight conjecture: This gives information on the Euler characteristic of a simplicial complex of a group.
Keywords
Literature
- Quillen 1978
- Rossi 2023