  After introducing the notion of uniform integrality of critical values of the
Rankin-Selberg $L$-functions for $\mathrm{GL}_{n}\times \mathrm{GL}_{n-1}$, we
study it when the base field is totally imaginary. For this purpose, we adopt
specific models of highest weight representations of the general linear groups,
construct the Eichler-Shimura classes for $\mathrm{GL}_{n}$ and
$\mathrm{GL}_{n-1}$ in an explicit manner, and then evaluate the cohomological
cup product of them, by making the best use of Gel'fand-Tsetlin basis.
