It is shown that if $λ$ is a multiple of a fundamental weight of $\mathfrak{sl}_k$, the lower global basis of the irreducible $U_q(\mathfrak{sl}_k)$-representation $V^λ$ with highest weight $λ$ comprises the disjoint union of the lower global bases of the irreducible $U_q(\mathfrak{sl}_{k-1})$-representations appearing in the decomposition of the restriction of $V^λ$ to $U_q(\mathfrak{sl}_{k-1})$. Rhoades's description of the action of the long cycle on the dual canonical basis of $V^λ$ is then deduced from Berenstein--Zelevinsky's description of the action of the long element. This yields a short proof of Rhoades's result on tableaux fixed under promotion which directly relates it to Stembridge's result on tableaux fixed under evacuation.