Molien series and low-degree invariants for a natural action of SO(3)wrZ2

Abstract

We investigate the invariants of the 25-dimensional real representation of the group SO(3)wr2 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 R3 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)wrZ2 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 = 1453926048 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.

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