Let $G$ be an algebraic group and $W$ denote the underlying affine Weyl group. I will label representations of the principal block of $G$ by their alcoves, which in turn I label by the corresponding element of the Weyl group $(W,S)$.
Given $w, y\in W$ Carter-Payne construct a homomorphism from $\Delta(w)$ to $\Delta(y)$ providing $w<y$ in the Bruhat order and $\ell(w)+1=\ell(y)=\ell$.
My question (and it should be silly and easy hopefully) is this: given a reduced expression $\underline{y}$ for $y$, how can I tell if a subexpression $\underline{w}$ is of length $\ell-1$? My suspicion is that the answer is as follows:
Fix $\underline{y}=s_1s_2\dots s_p$ with $s_i\in S$. A subexpression $\underline{w}=s_1s_2\dots s_{k-1}\widehat{s}_k s_{k+1}\dots s_\ell$ is reduced of length $\ell-1$ if and only if $s_k$ is the first or last occurrence of the generator $s_k$ in the expression $\underline{y}$.
I can prove this for type $A$, but I'm not really a dab hand at other Weyl groups. My feeling is that this should be true and well known, but I can't find a reference.
