definition of the pseudoscalar, pseudovector, and pseudotensor

I am fully obsessed with geometry these days. In particular, I am obsessed with the point that scalars aren't just numbers, but rather numbers that don't depend on your choice of coordinate system. Similarly, vectors aren't just things with a magnitude and a direction: They are things with a magnitude and a direction which are coordinate free, or which have a stable direction and magnitude no matter what coordinate system you choose. Thus, for example, the unit vectors defining the x, y, and z directions of your coordinate system are not really vectors at all. But the acceleration due to gravity right here is a vector.

But there are pseudo- quantities. For example, the angular momentum isn't exactly a vector; it is a pseudo-vector: It's magnitude and direction doesn't depend on the orientation (or translation) of the coordinate system, but its direction does depend on the handedness of the coordinate system. Thus there are pseudoscalar, pseudovector, and pseudotensors in addition to scalars, vectors, and tensors. Today Soledad Villar (JHU) wrote definitions for these in the paper we are drafting. It isn't trivial, because we want a notation that is agnostic about the group operator and the thing it is operating on.