Even partitions in plethysms

We prove that for all natural numbers k,n,d with k⩽d and every partition λ of size kn with at most k parts there exists an irreducible GLd(C)-representation of highest weight 2λ in the plethysm Symk(Sym2nCd). This gives an affirmative answer to a conjecture by Weintraub [Steven H. Weintraub, Some observations on plethysms, J. Algebra 129 (1) (1990) 103–114]. Our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory.