Affine standard Lyndon words

TL;DR

This paper generalizes properties of affine standard Lyndon words across all types, explores their structure, and offers computational tools for their efficient calculation, extending previous results in type A.

Contribution

It establishes convexity and monotonicity for affine standard Lyndon words in all types and provides computational methods for exceptional types.

Findings

Convexity and monotonicity proven for all types

Partial results on imaginary standard Lyndon words

Computer code for efficient computation in exceptional types

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.