Date: Monday, June 8, 2020
\nTime: 13:30 \u2013 14:30<\/p>\n
This is an online seminar. To obtain the event link, please send a message to o matthew.gelvin@bilkent.edu.tr<\/p>\n
Abstract: Fix a prime p. The fusion system of a finite group G with respect to a Sylow p-subgroup S of G is the category F_S(G) whose objects are the subgroups of S, and whose morphisms are the homomorphisms induced by conjugation in G. More generally, an abstract fusion system over a p-group S is a category F whose objects are the subgroups of S and whose morphisms are injective homomorphisms between the subgroups that satisfy certain axioms motivated by the Sylow theorems.<\/p>\n
The geometric realization |F| of a fusion system F is (roughly) the cell complex with one vertex for each object in F, one edge for each morphism (attached to its source and target vertices), one 2-simplex for each commutative triangle of morphisms, etc. The space |F| itself is contractible (as is the realization of every category with initial object), and hence not very interesting to an algebraic topologist.<\/p>\n
However, the realizations of certain full subcategories of F, and in particular their fundamental groups, do have important applications. For example, when F^c \u2286 F denotes the full subcategory of F-centric subgroups of S (very roughly, those that contain their centralizers in S), the group \u03c0_1(|F^c|) can be used to classify certain fusion subsystems of F. More recently, when F = F_S(G), for finite G and S \u2208 Syl_p(G), and F^\u2217 \u2286 F is the full subcategory whose objects are the nontrivial subgroups 1 < P \u2264 S, then \u03c0_1(|F^\u2217|) plays a key role in work by Grodal to describe the group of \u201cSylow trivial\u201d kG-modules when k is a field of characteristic p.<\/p>\n
We will describe these applications, and then give some examples of calculations that have been made of these and other fundamental groups.<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
Date: Monday, June 8, 2020 Time: 13:30 \u2013 14:30 This is an online seminar. To obtain the event link, please send a message to o matthew.gelvin@bilkent.edu.tr Abstract: Fix a prime p. The fusion system of a finite group G with respect to a Sylow p-subgroup S of G is the [\u2026] <\/p>\n