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Decomposing symmetric powers of modular representations of elementary abelian groups

Let G be an elementary abelian p-group of order q, V a 2-dimensional faithful modular representation of G, and W the dual of some indecomposable symmetric power of V. We consider the symmetric algebra on W, in particular showing that all its summands are projective relative to certain subgroups of G in high enough degrees. Our results generalise results of Almkvist and Fossum for cyclic p-groups. We also give some applications to invariant theory.