Tensor Conventions: V^*⊗V^*⊗V (1,2) vs (2,1)

[fresh_42]

How do physicists call a tensor of $V^* \otimes V^* \otimes V$, (1,2) or (2,1)?
And which part do they call contravariant and which covariant?

I'm just not sure, whether the mathematical definition of funktors apply to the usances in physics.
(LUP - tensor)

 

[Orodruin]

I would say (1,2) - one in V and two in its dual. I would not call the tensor itself covariant or contravariant. I would say that the components transform co- or contravariantly. In this case, the components would be of the form $T^a_{bc}$ with the $a$ being labelled as a contravariant index whereas $bc$ would be labelled covariant.

 

[fresh_42]

Thanks, that was my understanding, too. And, yes, that was my problem: the distinction between "transforms as" which refers to the behavior in a change of coordinates and the property itself, because mathematically the dual $bc$ part changes direction of morphisms and thus should have been contra instead of co. So the point is, that mathematicians use these terms as a property of functors whereas physicists refer to coordinates w.r.t. the same object.

I hope I finally got it now.