  We prove a combinatorial rule for a complete decomposition, in terms of
Langlands parameters, for representations of p-adic $GL_n$ that appear as
parabolic induction from a large family (ladder representations). Our rule
obviates the need for computation of Kazhdan-Lusztig polynomials in these
cases, and settles a conjecture posed by Lapid.
  These results are transferrable into various type A frameworks, such as the
decomposition of convolution products of homogeneous KLR-algebra modules, or
tensor products of snake modules over quantum affine algebras.
  The method of proof applies a quantization of the problem into a question on
Lusztig's dual canonical basis and its embedding into a quantum shuffle
algebra, while computing numeric invariants which are new to the p-adic
setting.
