On the symmetric and exterior powers of Young permutation modules

Let $n$ be a positive integer and $\lambda$ be a partition of $n$. Let $M^\lambda$ be the Young permutation module labelled by $\lambda$. In this paper, we study symmetric and exterior powers of $M^\lambda$ in positive characteristic case. We determine the symmetric and exterior powers of $M^\lambda$ that are projective. All the indecomposable exterior powers of $M^\lambda$ 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 $M^\lambda$. We end by parameterizing all the Scott modules that are isomorphic to direct summands of the symmetric or exterior square of $M^\lambda$ and determining their corresponding multiplicities explicitly.