  This paper is concerned with the subduction problem of type A quantum
Iwahori-Hecke algebras $\mathbb{C} \mathbf{H}(\mathfrak{S}_f,q^2)$ with a real
deformation parameter $q$, i.e. the problem of decomposing irreducible
representations of such algebras as direct sum of irreducible representations
of the subalgebras $\mathbb{C}\mathbf{H}(\mathfrak{S}_{f_1}, q^2) \times
\mathbb{C}\mathbf{H}(\mathfrak{S}_{f_2}, q^2)$, with $f_1 + f_2 = f$. After
giving a suitable combinatorial description for the subduction issue, we
provide a selection rule, based on the Richardson-Littlewood criterion, which
allows to determine the vanishing coupling coefficients between standard basis
vectors for such representations, and we also present an equivariance condition
for the subduction coefficients. Such results extend those ones corresponding
to the subduction problem in symmetric group algebras $\mathbb{C}\mathfrak{S}_f
\downarrow \mathbb{C}\mathfrak{S}_{f_1} \times \mathbb{C} \mathfrak{S}_{f_2}$
which are obtained by $q$ approaching the value 1.
