  Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length
representations of the sequence of p-adic groups $\{GL_n(F)\}_{n=0}^\infty$,
with multiplication defined through parabolic induction. We study the problem
of the decomposition of products of irreducible representations in
$\mathcal{R}$.
  We obtain a necessary condition on irreducible factors of a given product by
introducing a width invariant. Width $1$ representations form the previously
studied class of ladder representations.
  We later focus on the case of a product of two ladder representations, for
which we establish that all irreducible factors appear with multiplicity one.
  Finally, we propose a general rule for the composition series of a product of
two ladder representations and prove its validity for cases in which the
irreducible factors correspond to smooth Schubert varieties.
