  It is shown that if $\lambda$ is a multiple of a fundamental weight of
$\mathfrak{sl}_k$, the lower global basis of the irreducible
$U_q(\mathfrak{sl}_k)$-representation $V^{\lambda}$ with highest weight
$\lambda$ 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 $V^{\lambda}$ to
$U_q(\mathfrak{sl}_{k-1})$. Rhoades's description of the action of the long
cycle on the dual canonical basis of $V^{\lambda}$ 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.
