The strong Centre Conjecture: an invariant theory approach

TL;DR

This paper introduces an invariant theory approach to a strengthened form of Tits' Centre Conjecture for spherical buildings, generalizing Kempf's result to prove the conjecture in specific cases.

Contribution

It generalizes Kempf's invariant theory concepts to spherical buildings, providing a new framework to approach the Centre Conjecture.

Findings

Proved the Centre Conjecture in some special cases.

Recaptured the conjecture entirely through the generalized notion of a state.

Established a new invariant theory approach for spherical buildings.

Abstract

The aim of this paper is to describe an approach to a a strengthened form of J. Tits' Centre Conjecture for spherical buildings. This is accomplished by generalizing a fundamental result of G. R. Kempf from Geometric Invariant Theory and interpreting this generalization in the context of spherical buildings. We are able to recapture the conjecture entirely in terms of our generalization of Kempf's notion of a state. We demonstrate the utility of this approach by proving the Centre Conjecture in some special cases.