The composition of two Schur functors, so called plethysm, in general is a reducible representation. The problem of decomposing it into irreducible components is well-known and hard.
We will present the results of a joint work with Thomas Kahle, where we provide formulas for certain plethysms by relating them to convex polyhedral geometry. Our algorithmic approach relies on the Barvinok software used to obtain quasipolynomials counting lattice points in fibers of a projection of a polyhedral cone.
In particular, we provide explicit formulas for $S^\mu(S^k)$ where $\mu$ is any partition of 4 or 5 and k is a parameter. We also prove asymptotic formulas conjectured by Howe in case when $\mu$ is any partition.