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

TL;DR

This paper advances the algebraic understanding of Erd ext{o}s-Ko-Rado sets of flags in spherical buildings by analyzing eigenspaces and providing combinatorial descriptions, aiding classification of maximal sets.

Contribution

It describes eigenspaces for the smallest eigenvalue of opposition graphs and offers a combinatorial framework for classifying maximal Erd ext{o}s-Ko-Rado sets of flags.

Findings

Determined multiplicities of eigenspaces.

Provided combinatorial descriptions of spanning sets.

Facilitated classification of maximal EKR-sets for certain types.

Abstract

We continue our investigation of Erd\H{o}s-Ko-Rado (EKR) sets of flags in spherical buildings. In previous work, we used the theory of buildings and Iwahori-Hecke algebras to obtain upper bounds on their size. As the next step towards the classification of the maximal EKR-sets, we describe the eigenspaces for the smallest eigenvalue of the opposition graphs. We determine their multiplicity and provide a combinatorial description of spanning sets of these subspaces, from which a complete description of the maximal Erd\H{o}s-Ko-Rado sets of flags may potentially be found. This was recently shown to be possible for type $A_n$, $n$ odd, by Heering, Lansdown, and the last author by making use of the current work.