  We give an introduction to some of the recent ideas that go under the name
"geometric complexity theory". We first sketch the proof of the known upper and
lower bounds for the determinantal complexity of the permanent. We then
introduce the concept of a representation theoretic obstruction, which has
close links to algebraic combinatorics, and we explain some of the insights
gained so far. In particular, we address very recent insights on the complexity
of testing the positivity of Kronecker coefficients. We also briefly discuss
the related asymptotic version of this question.
