Minuscule weights and Chevalley groups

TL;DR

This paper extends Lusztig's simplified construction of Chevalley groups to non-adjoint types by utilizing Jantzen's models of minuscule highest weight representations, removing the need for sign choices.

Contribution

It introduces a new method for constructing Chevalley groups of arbitrary type using minuscule representations, generalizing previous approaches that required sign choices.

Findings

Simplifies Chevalley group construction for non-adjoint types

Uses Jantzen's explicit models of minuscule representations

Removes sign ambiguity in the construction process

Abstract

The traditional construction of Chevalley groups relies on the choice of certain signs for a Chevalley basis of the underlying Lie algebra~$\mathfrak{g}$. Recently, Lusztig simplified this construction for groups of adjoint type by using the "canonical basis" of the adjoint representation of~$\mathfrak{g}$, in particular, no choices of signs are required. The purpose of this note is to extend this to Chevalley groups which are not necessarily of adjoint type, using Jantzen's explicit models of the minuscule highest weight representations of~$\mathfrak{g}$.