Quantum invariants for decomposition problems in type A rings of representations

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