Tensor products of Young modules

We apply the Brauer construction to tensor products of trivial-source modules. If U and V are trivial-source kG-modules with vertices P and Q respectively, we describe a family X of subgroups of G depending only on P and Q, such that for every subgroup H in X, there exists a direct summand M of U⊗V with vertex H, and a correspondence for such summands. We apply this to the special case of Young modules for symmetric groups, in particular deriving similar results concerning Young vertices, and some reduction theorems for multiplicities of Young modules in a tensor product of Young modules. Some new lower bounds on certain Cartan numbers for Schur algebras are obtained as an application of the results on Young modules.