Consider the symmetric group $W=S_3$, and let $A=k[V]$ be the regular functions on its reflection representation $V$. Consider, for a word $w\in W$ the quotients
$$A_w=\frac{A\otimes A}{\prod_{w'\leq w}(w'a\otimes 1-1\otimes a)}$$
Denote by $A_w^+$ the positively graded part of its $s\times 1$-invariants. Then it is claimed that the multiplication $A\otimes_{A^s}A_w^+ \longrightarrow A_w$ is an isomorphism if $sw<w$.
I am not able to see why this holds. For $w=s$ I can calculate this manually, but I see no way to generalise this. Any hints? Why do I need $sw<w$?