On the symmetric and exterior powers of Young permutation modules

摘要

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. (C) 2020 Elsevier Inc. All rights reserved.