Algorithm for computing the number of independent real parameters in the phonon dynamical matrix

The effects of time-reversal symmetry in reducing the maximum number $\mathfrak{M}$ of real parameters in the phonon dynamical matrix at fixed $\overrightarrow{\mathrm{k}}$ are taken into account to yield the following result: either (I) -$\overrightarrow{\mathrm{k}}$ is not in the star of $\overrightarrow{\mathrm{k}}$ in which case ${\mathfrak{M}}^{\mathrm{I}}=\Sigma r^{}{n}_r^2$, or (II) it is, in which case ${\mathfrak{M}}^{\mathrm{II}}=\Sigma {r\in \left(\mathrm{a}\right)}^{}{n}_r\frac{(n_r+1)}{2}+\Sigma {r\in \left(\mathrm{b}\right)}^{}\frac{n_r^2}{2}+\Sigma {r\in \left(\mathrm{c}\right)}^{}{n}_r\frac{(n_r-1)}{2}$. Here, $n_r$ denotes the number of times the $r$ th irreducible representation of the little group $G^{\overrightarrow{\mathrm{k}}}$ occurs in the reducible representation ${\Delta }^{\overrightarrow{\mathrm{k}}}$ at wave vector $\overrightarrow{\mathrm{k}}$. ${\Delta }^{\overrightarrow{\mathrm{k}}}$ is of dimension $3n$, where $n$ is the number of atoms per unit cell. In the expression for ${\mathfrak{M}}^{\mathrm{II}}$, one sums over all $r$ in cases (a), (b), and (c), respectively, where (a), (b), and (c) refer to the three cases of Wigner in the Herring criterion. The analysis is also applied to compute a lower upper bound ${\mathfrak{M}}^{\prime }$ within the rigid-molecule model as discussed group theoretically in the context of inelastic-neutron-scattering experiments by Casella and Trevino.