Molien series and low-degree invariants for a natural action of SO(3)wr Z_2

We investigate the invariants of the 25-dimensional real representation of the group SO(3) \wr Z_2 given by the left and right actions of SO(3) on 5 × 5 matrices together with matrix transposition; the action on column vectors is the irreducible five-dimensional representation of SO(3) . The 25-dimensional representation arises naturally in the study of nematic liquid crystals, where the second-rank orientational order parameters of a molecule are represented by a symmetric 3 × 3 traceless symmetric matrix, and where a rigid rotation in R^3 induces a linear transformation of this space of matrices. The entropy contribution to a free energy density function in this context turns out to have SO(3) \wr Z_2 symmetry. Although it is unrealistic to expect to describe the complete algebraic structure of the ring of invariants, we are able to calculate as a rational function the Molien series that gives the number of linearly independent invariants at each homogeneous degree. The form of the function indicates a basis of 19 primary invariants and suggests there are N=1 453 926 048 linearly independent secondary invariants; we prove that their number is an integer multiple of N/4. The algebraic structure of invariants up to degree 4 is investigated in detail.