An algebraic approach to Erd\H{o}s-Ko-Rado sets of flags in spherical buildings

TL;DR

This paper introduces an algebraic framework using spherical buildings and Iwahori-Hecke algebras to establish sharp bounds on Erd ext{"o}s-Ko-Rado sets of flags in projective and polar spaces, generalizing previous results.

Contribution

It develops a novel algebraic approach combining building theory and Iwahori-Hecke algebras to derive upper bounds for EKR-sets, extending prior bounds in classical geometries.

Findings

Established sharp upper bounds for EKR-sets of flags

Reproved and generalized previous bounds in projective and polar spaces

Connected building theory with algebraic combinatorics techniques

Abstract

In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bounds for EKR-sets of flags. In this framework, we can reprove and generalize previous upper bounds for EKR-problems in projective and polar spaces. The bounds are obtained by the application of the Delsarte-Hoffman coclique bound to the opposition graph. The computation of its eigenvalues is due to earlier work by Andries Brouwer and an explicit algorithm is worked out. For the classical geometries, the execution of this algorithm boils down to elementary combinatorics. Connections to building theory, Iwahori-Hecke algebras, classical groups and diagram geometries are briefly discussed. Several open problems are posed throughout and at the end.