Chains of affine standard Lyndon words

TL;DR

This paper proves the periodicity of chains of affine standard Lyndon words across all types and establishes precise bounds, extending previous results from type A.

Contribution

It generalizes the periodicity results of affine standard Lyndon words to all types and introduces a new approach using root system polarization.

Findings

Established periodicity of affine standard Lyndon word chains in all types.

Determined tight bounds on the periodicity.

Extended type A results to all types.

Abstract

In this note, we establish the periodicity of chains of affine standard Lyndon words in all types and determine tight bounds on that periodicity, greatly generalizing the $A$-type results of arXiv:2305.16299. Our approach crucially utilizes the convexity and monotonicity of arXiv:2505.15432 together with the new idea to consider the polarization of the root system given by increasing and decreasing chains.