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Molien series and low-degree invariants for a natural action of ${\bf SO}(3)\wr{\bf Z}_2$

We investigate the invariants of the $25$-dimensional real representation of the group ${\bf SO}(3)\wr{\bf Z}_2$ given by the left and right actions of ${\bf SO}(3)$ on $5\times 5$ matrices together with matrix transposition; the action on column vectors is the irreducible $5$-dimensional representation of ${\bf SO}(3)$. 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 $3\times3$ traceless symmetric matrix, and where a rigid rotation in ${\bf R}^3$ 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 ${\bf SO}(3)\wr{\bf Z}_2$ symmetry. Although it is unrealistic to expect to describe the complete algebraic structure of the ring of invariants, we are able to calculate the Molien series giving the number of linearly independent invariants at each homogeneous degree, and to express this as a rational function indicating the degrees of invariant polynomials that constitute a basis of 19 primary invariants. The algebra of invariants up to degree 4 is investigated in detail.