Let $a$ and $b$ be two generators in a Coxeter group which do not commute. Is it possible for $ab$ to be equal to a product of generators where all instances of $b$ come before all instances of $a$?


I've tried coming up with invariants that are preserved after applying the conditions of a Coxeter group to a string of generators, but the fact that one can insert either $aa$ or $bb$ at any point in the string has made this complicated. Particularly, the more general claim that a string with all $a$s before $b$s can't be turned into one with all $b$s before $a$s turns out to be false, since for instance $aab=baa$.



Swapping non-commuting generators in Coxeter group
The answer is no. The Deletion Condition says that any expression in the generators of a Coxeter group contains a reduced expression for the same element as a subexpression. Since $a$ and $b$ don't commute, the unique reduced expression for $ab$ is $ab$, which must therefore occur as a subexpression of any longer expression that evaluates to $ab$, i.e. any such expression must contain an $a$ before a $b$.


