(Joint Algebra-Data Seminar)

Invariant polynomials have been recently shown to be useful for finding maximally entangled quantum systems for small numbers of particles.  Invariant rings for spaces of tensors are particularly nice: they are finitely generated, and Cohen-Macaulay, and in the tame cases (the set of orbits depending on finitely many parameters) they are even polynomial rings. We can compute their Hilbert series via computations with finite groups. But what if we want to really get our hands on computations with invariants? Schur-Weyl duality and Young symmetrizers provide a way to access and evaluate these functions at least in low degrees. The first case of qubits with wild orbit type is with 5 qubits, and in this case it might be too difficult to compute all generators of the invariant ring. Instead, we could be happy with a set of primary invariants. I will discuss our implementations of these invariant theory computations in Macaulay2 and attempts to find sets of primary invariants in this case.