The unique coclique extension property for apartments of buildings

TL;DR

This paper investigates the unique coclique extension property in Kneser graphs derived from objects in buildings of spherical type, focusing on specific conditions related to minuscule weights and simply laced diagrams.

Contribution

It establishes the conditions under which the Kneser graph of objects in a building has the unique coclique extension property, linking geometric and algebraic properties.

Findings

Kneser graph has the property when the representation has minuscule weight.

The property holds when the diagram is simply laced and the representation is adjoint.

Results connect building theory with graph extension properties.

Abstract

We show that the Kneser graph of objects of a fixed type in a building of spherical type has the unique coclique extension property when the corresponding representation has minuscule weight and also when the diagram is simply laced and the representation is adjoint.