Let $\Phi$ be the root system of type $A$. Let $\mathcal{A}$ be an alcove of the corresponding affine arrangement. The address (or Shi coordinates) of $\mathcal{A}$ is a function $k:\Phi^+ \rightarrow \mathbb{Z}$ satisfying $$k(\epsilon_i - \epsilon_j) < (\lambda, \epsilon_i - \epsilon_j) < k(\epsilon_i - \epsilon_j) + 1,$$ for all $\lambda \in \mathcal{A}$. (Here $(\cdot,\cdot)$ is the standard inner product on $\mathbb{R}^n$.)
Let $\leq_R$ denote the right weak order on $\widetilde{A}_n$. For $w \in \widetilde{A}_n$, let $k_w$ denote the address of the alcove corresponding to $w$. Weak order is characterized by $$u \leq_R v \iff |k_u| \leq |k_v| \text{ and } \text{sgn}(k_u) = \text{sgn}(k_v),$$ where the signs are regarded as equal whenever $k_u(\epsilon_i - \epsilon_j) = 0$.
I'm looking for a reference for this characterization.
One can prove this in general by applying Theorem 4.5 of Humphreys' Reflection groups and Coxeter groups. Also, connecting Theorem 4.1 of Shi's On two presentations of the affine Weyl groups of classical types with Theorem 5.3 of Bj$\ddot{\text{o}}$rner and Brenti's Affine permutations of type A works. Is there something more direct in the literature?
