2025/05/20 by Jan De Beule, Philipp Heering, De Beule, Jan +5 · 1 citation
Mathematics · #05C35 #05C50 #05C69 #Advanced Operator Algebra Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2505.14322
openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Erdős-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 An, 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 2A4n-3.