The largest sets of non-opposite chambers in spherical buildings of type $B$

TL;DR

This paper studies large families of non-opposite chambers in spherical buildings of type B, extending extremal combinatorics and algebraic techniques to classify maximal families across various cases.

Contribution

It introduces a uniform method based on antidesigns to classify maximal non-opposite chamber families in type B spherical buildings, generalizing previous results.

Findings

Established upper bounds for family sizes in type B buildings.

Classified maximal families of chambers in all cases except one.

Extended algebraic and combinatorial techniques to new building types.

Abstract

The investigation into large families of non-opposite flags in finite spherical buildings has been a recent addition to a long line of research in extremal combinatorics, extending classical results in vector and polar spaces. This line of research falls under the umbrella of Erd\H{o}s-Ko-Rado (EKR) problems, but poses some extra difficulty on the algebraic level compared to aforementioned classical results. From the building theory point of view, it can be seen as a variation of the center conjecture for spherical buildings due to Tits, where we replace the convexity assumption by a maximality condition. In previous work, general upper bounds on the size of families of non-opposite flags were obtained by applying eigenvalue and representation-theoretic techniques to the Iwahori-Hecke algebras of non-exceptional buildings. More recently, the classification of families reaching this upper bound in type $A_n$, for $n$ odd, was accomplished by Heering, Lansdown, and Metsch. For buildings of type $B$, the corresponding Iwahori-Hecke algebra is more complicated and depends non-trivially on the type and rank of the underlying polar space. Nevertheless, we are able to find a uniform method based on antidesigns and obtain classification results for chambers (i.e.\ maximal flags) in all cases, except type $^2A_{4n-3}$.