# Reduced expression and Bruhat order

For $$n\ge 3$$. Let $$s_1\cdots s_n$$ be a reduced expression of $$x$$. Suppose $$s_1\cdots s_{n-1}\le w$$ and $$s_2\cdots s_{n}\le w$$.

Does this imply $$x\le w$$?

• Have you tried looking at some small Coxeter groups to see if this is true? – user44191 Aug 13 '19 at 3:12

Let $$W = S_5$$ (as a side note, we could let $$W = S_3 \times S_2$$). Let $$w = (132)(45), x = (123)(45) \in W$$; we write $$x = (12)(45)(23)$$ as a reduced expression. Then the subexpressions written in the question are $$(12)(45)$$ and $$(45)(23) = (23)(45)$$. The reduced expressions for $$w$$ are $$(23)(12)(45), (23)(45)(12), (45)(23)(12)$$; for each of those, the appearance of both $$(12)(45) = (45)(12), (45)(23) = (23)(45)$$ as outputs for subexpressions is clear, so $$(12)(45), (23)(45) \leq w$$, satisfying the hypothesis of the theorem, but $$x \nleq w$$, so the answer to the question is no.