Affine standard Lyndon words
Corbet Elkins, Alexander Tsymbaliuk
TL;DR
The paper establishes convexity and monotonicity for affine standard Lyndon words across all Lie types, generalizing known A-type results. It develops a framework of extended root sets, flags, and W/O-sets to analyze bracketing without relying on explicit closed formulas, and proves the convexity/monotonicity theorems that underpin the affine Lyndon word structure. It also provides a conjecture for the structure of imaginary affine standard Lyndon words and proves it in significant cases (notably when the smallest simple root occurs once in the imaginary delta), recovering the A-type behavior in general. An explicit computational appendix accompanies the theoretical results, enabling efficient calculation of affine standard Lyndon words in exceptional types and supporting the verification of conjectures.
Abstract
In this note, we establish the convexity and monotonicity for affine standard Lyndon words in all types, generalizing the $A$-type results of arXiv:2305.16299. We also derive partial results on the structure of imaginary standard Lyndon words and present a conjecture for their general form. Additionally, we provide computer code in Appendix which, in particular, allows to efficiently compute affine standard Lyndon words in exceptional types for all orders.