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