Abstract: Assume $\mathscr{F}$ to be a field of characteristic zero,
and $q\in \mathscr{F}$ to be a root of unity. With $\mathscr F$ as
the ground field and $q$ as the quantum parameter, let
$\mathsf{s}_q(n)$ be the restricted quantum symmetric algebra of
rank $n$, and $\Wedge_q(n)$ be the quantum exterior algebra of rank
$n$. By [6], the homogenous components of both $\mathsf{s}_q(n)$ and
$\Wedge_q(n)$ are simple $U_q(\mathfrak{gl}_n)$-modules. In this
paper, we decompose the tensor product of any homogenous component
of $\mathsf{s}_q(n)$ with any homogenous component of $\Wedge_q(n)$
into direct sum of indecomposable modules.

Key words: quantum group, restricted quantum symmetric algebra, quantum exterior algebra, Krull--Schmidt decomposition