Support of positive Artin words and longest lifts

Each defining positive braid relation replaces an alternating word on two distinct generators by another word with exactly the same two generator indices. A finite sequence of such replacements inside words therefore preserves the set of indices. By the presented-monoid identification in [F1], all positive representatives of an element are connected by these replacements. Support is consequently well defined. Concatenation gives the union formula. An element represented using only $I$ has support contained in $I$; conversely any positive representative of an element whose support is contained in $I$ uses only $I$.

By [F3], $(W_I,I)$ is a spherical Coxeter system with intrinsic length equal to ambient length. Its longest element has a reduced expression using only $I$. For each $i\in I$, its generator lift is a simple and hence a left divisor of the restricted longest lift by [F1]. The map sending the restricted positive generators to the ambient generators respects every defining braid relation, so it maps this positive factorization into the ambient monoid. By the length agreement in [F3], the restricted reduced expression is ambient reduced; [F2] therefore identifies the image of its lift with the stated ${\mathrm{\Delta }}_I$. Thus $\mathrm{supp}({\mathrm{\Delta }}_I)\subseteq I$, while every $i\in I$ occurs in that support by the mapped factorization and step 1.1. Its support is exactly $I$. For $I=\mathrm{\varnothing }$ the lift is the empty product.