  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.
