Lines in geometries associated to finite buildings

by Sira Busch (Universität Münster)

Monday 24 Feb 2025, 14:00 → 15:00 Europe/Brussels

B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

 

Suppose ΔΔ is an irreducible, thick, finite, spherical building of rank greater one. We call a panel ss-thick, if it is contained in precisely s+1s+1 chambers. Suppose the panels of cotype {i}{i} in ΔΔ are ss-thick. Every set of ss vertices of type {i}{i} admits a common opposite vertex. This not true for s+1s+1 vertices of type {i}{i}. Hence the question: When is it the case that we can find a common opposite vertex in ΔΔ to s+1s+1 given vertices of type {i}{i}? Answering this question is an ongoing project of Hendrik Van Maldeghem and myself that we already solved for the classical case. This question arose while working on constructions in finite Lie incidence geometries that resulted in us, together with Jeroen Schillewaert, determining all Levi subgroups of parabolic subgroups of groups of Lie type related to thick, irreducible, spherical buildings of simply laced type. Viewed in another way, answering this question lays a basis for investigating the analogues of blocking sets in Lie incidence geometries. In my talk I would like to explain our project and how it connects group theory, incidence geometry and combinatorics.