Inspired by questions in computational complexity theory, in particular by the geometric complexity theory program by Mulmuley and Sohoni, this talk is a report on the following 3 results that are of independent interest in algebraic geometry and representation theory.

The famous Foulkes conjecture (1950) is a conjecture about the inequality of plethysm coefficients in Sym^a Sym^b V and Sym^b Sym^a V. The conjecture is known for a <= 4. We prove the case a=5.

We prove and generalize Weintraub's conjecture (1990) that states that all even partitions occur in the plethysm Sym^d Sym^n V for n even.

Related to the fundamental positivity questions in representation theory is the following question: What is the computational complexity of deciding the positivity of the Kronecker coefficient? We prove that this is NP-hard by giving a combinatorial interpretation of the Kronecker coefficient in a hard subcase.