Generic direct summands of tensor products for simple algebraic groups and quantum groups

Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of positive characteristic or a quantum group at a root of unity. We define new classes of indecomposable $\mathbf{G}$-modules, which we call generic direct summands of tensor products because they appear generically in Krull-Schmidt decompositions of tensor products of simple $\mathbf{G}$-modules and of Weyl modules. We establish a Steinberg-Lusztig tensor product theorem for generic direct summands of tensor products of simple $\mathbf{G}$-modules and provide examples of generic direct summands for $\mathbf{G}$ of type ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$.