On the labelling of characters of Weyl groups of type $F_4$

TL;DR

This paper examines two different conventions for labeling irreducible characters of Weyl groups of type F4, highlighting issues and their impact on related tables, and identifies an error in Springer correspondence tables for type E7.

Contribution

It clarifies the differences between labeling conventions for F4 Weyl groups and uncovers an error in existing Springer correspondence tables for E7.

Findings

Identified issues with two labeling conventions for F4 Weyl groups.

Demonstrated the impact of these issues on unipotent character tables.

Discovered an error in Spaltenstein's Springer correspondence tables for E7.

Abstract

In the literature on finite groups of Lie type, there exist two different conventions about the labelling of the irreducible characters of Weyl groups of type~$F_4$. We point out some issues concerning these two conventions and their effect on tables about unipotent characters or the Springer correspondence. Using experiments related to these issues with the computer algebra system {\sf CHEVIE}, we spotted an error in Spaltenstein's tables for the generalised Springer correspondence in type~$E_7$.