On the symmetric and exterior powers of Young permutation modules

Abstract Let n be a positive integer and λ be a partition of n. Let M λ be the Young permutation module labelled by λ. In this paper, we study symmetric and exterior powers of M λ in positive characteristic case. We determine the symmetric and exterior powers of M λ that are projective. All the indecomposable exterior powers of M λ are also classified. We then prove some results for indecomposable direct summands that have the largest complexity in direct sum decompositions of some symmetric and exterior powers of M λ . We end by parameterizing all the Scott modules that are isomorphic to direct summands of the symmetric or exterior square of M λ and determining their corresponding multiplicities explicitly.