Correction of the Lusztig-Williamson Billiards Conjecture

TL;DR

This paper introduces a new algorithm for calculating tilting characters in algebraic groups, revealing that the Lusztig-Williamson Billiards Conjecture requires correction based on new computational evidence.

Contribution

The paper presents a novel algorithm for computing tilting characters and corrects the Lusztig-Williamson Billiards Conjecture based on these new calculations.

Findings

New calculations for $SL_3$, $SP_4$, $G_2$, $SL_4$

Evidence that the Lusztig-Williamson Billiards Conjecture is incorrect

Proposed correction to the conjecture

Abstract

A new algorithm allows us to calculate many new tilting characters for $SL_3$, $SP_4$, $G_2$, $SL_4$ and potentially many other groups. These calculations show that the Lusztig-Williamson Billiards Conjecture needs to be corrected. In this paper we present the new results calculated for $SL_3$ and a correction of the conjecture.