Neumann's principle (group theory applied to crystals), Seebeck tensor

[fluidistic]

I am reading about symmetries in crystals, and my knowledge in the field of group theory is almost nill. I am reading that, in the worst case, the electrical and thermal conductivity tensors can possess, at maximum, 6 different entries rather than 9, thanks to Neumann's principle which states (in words): "The symmetry of any physical property of a crystal must include the symmetry elements of the point group of the crystal.", ref. "properies of materials" by Newnham. Intuitively, from what I understand, is that if the crystal possesses symmetries, then its physical properties must also, at least, possess these symmetries (although they can have more symmetries than the crystal itself).

But then, it is mentioned that the Seebeck tensor can have as many as 9 different entries, something impossible for the other transport properties. How is that possible, how does that not violate Neumann's principle?

 

[Valerie Park]

The answer is that the Seebeck tensor is an antisymmetric tensor, meaning that it changes sign when its indices are swapped. This means that the symmetry elements of the point group of the crystal do not have to apply in order for the Seebeck tensor to have 9 different entries. The antisymmetry of the tensor allows for the separate consideration of the different components of the tensor, thereby breaking the symmetry of the point group and allowing for up to nine different entries in the Seebeck tensor.