# Swapping non-commuting generators in Coxeter group

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 $$aab=baa$$.

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 $$b$$.

• Isn't the "deletion condition" a theorem rather than a "condition"?
– YCor
May 13 '21 at 7:27
• @YCor: I agree that the terminology is perhaps strange, but that's how it is in the book by Humphreys. May 13 '21 at 9:03
• Is there any standard reading for this topic? May 13 '21 at 15:19
• @URL: Björner and Brenti's "Combinatorics of Coxeter Groups" talks about the Deletion Property. Look at Section 1.4. (By the way, Björner and Brenti call it a "property", not a "condition".) May 14 '21 at 0:58