## Lyndon words

Last update: 15 March 2013

## Lyndon words

The lexicographic order on words is given by ${f}_{1}<\cdots <{f}_{r}$ and $u A word is Lyndon if it is smaller than all its proper right factors.

Any word $u$ has a unique factorisation $u=l1 l2⋯lk with l1≥l2 ≥⋯≥lk Lyndon.$ (see [Lo, Thm 5.1.5] or [Re, Thm 4.9]).

If $\gamma =4{\alpha }_{1}^{\vee }+{\alpha }_{2}^{\vee }$, the words in the set ${\Gamma }^{\gamma }$ (displayed in their nonincreasing Lyndon factorisation) are $f1f1 f1f1 f2, (f1f1 f1f2) ⋅f1, (f1f1 f2) ⋅f1⋅ f1, (f1f2) ⋅f1⋅ f1⋅ f1, and f2⋅ f1⋅ f1⋅ f1⋅ f1.$

A word is good if there is a homogeneous element $a\in {U}_{q}{𝔫}^{-}$ such that $g$ is the maximal word appearing in $a$. The following proposition gives a characterisation of good words and good Lyndon words.

[Le, Prop 17, Prop 25] and [LR, Cor 2.5]

1. A word $g$ is good if and only if $g=l1⋯ lk with l1≥l2 ≥⋯≥lk good Lyndon words.$
2. Let ${R}^{+}$ be the set of positive roots and let ${ℛ}^{+}$ be the set of good Lyndon words. Then the map $R+ → ℛ+ β ↦ l(β) is a bijection,$ where $l(β) =max{ l(β1) l(β2) | β1, β2 ∈R+, β= β1+β2, l(β1)

## Notes and References

This page is partially based on joint work with P. Lalonde and A. Kleshchev.