If a space point group for a crystal is known, does this automatically define the elastic tensor symmetry of the material? What further implications can be found?
The crystallographic subgroups:
For example, the numbers of the elastic tensor parameters need not be known, but certain relations must exist in order for the energy of the crystal to be positive definite, which is a consequence of the structure's stability. One such combinations of relations for a hexagonally symmetric elastic tensor is that
$$C_{44} > 0, \,\, C_{11} > | C_{12} |, \,\,\,\,\left(C_{11} + C_{12} \right) C_{33} > 2 C_{13}^2$$
So for Zn, which is of the point group $P6_3/\mathrm{mmm}$, has hexagonal structure, this automatically ensures the above case? Is there a situation in which this is not true? It is obvious that ANY hexagonal structure should have the above case but is there a general method for finding the additional constraints for say the differences between $P6mm$ and $P\bar{6}m2$?