Enhanced Dynkin diagrams and Weyl orbits

TL;DR

This paper introduces enhanced Dynkin diagrams and bases for ADE root systems, enabling a positive correspondence between root system homomorphisms and basis homomorphisms, and clarifies the partial order of Weyl orbits.

Contribution

It proposes an enhanced basis and Dynkin diagram that improve understanding of root system homomorphisms and Weyl orbit relations for ADE systems.

Findings

Enhanced bases allow all root system homomorphisms to be derived from basis homomorphisms.

Enhanced Dynkin diagrams provide a clearer visualization of Weyl orbit partial orders.

New tools like mosets and core groups facilitate the analysis of ADE root systems.

Abstract

The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of R is, in general, not large enough to contain the diagrams of all subsystems of R, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in R. In this paper we consider only ADE root systems (that is, systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.