Cyclic symmetry of the scaled simplex

Let $\mathcal{Z}_{m}^{k}$ consist of the m ^k alcoves contained in the m-fold dilation of the fundamental alcove of the type A _k affine hyperplane arrangement. As the fundamental alcove has a cyclic symmetry of order k+1, so does $\mathcal{Z}_{m}^{k}$ . By bijectively exchanging the natural poset structure of $\mathcal{Z}_{m}^{k}$ for a natural cyclic action on a set of words, we prove that $(\mathcal{Z}_{m}^{k},\prod_{i=1}^{k} \frac{1-q^{m i}}{1-q^{i}},C_{k+1})$ exhibits the cyclic sieving phenomenon.