Terminology for (anti)symmetric tensors in characteristic 2

[Hurkyl]

When working over a field of characteristic not 2, or otherwise with modules over a ring where 2 is invertible, there is no ambiguity in what one means by symmetric or anti-symmetric rank 2 tensors. All of definitions of the anti-symmetric tensors

• The module of anti-symmetric tensors is the quotient of M \otimes M by the relations x \otimes y = -(y \otimes x).
• The module of anti-symmetric tensors is the quotient of M \otimes M by the relations x \otimes x = 0.
• The module of anti-symmetric tensors is the image of the anti-symmetrization operation x \otimes y \mapsto (1/2)(x \otimes y - y \otimes x) on M \otimes M
• The module of anti-symmetric tensors is the kernel of the symmetrization operation x \otimes y \mapsto (1/2)(x \otimes y + y \otimes x)

give the same module, and there is a standard notation for it: \wedge^2 M or M \wedge M.

Similarly for symmetric tensors.



When 2 is not invertible, the different definitions can give different groups. Is there standard terminology and notation for the various possibilities? The only one I'm aware of is that the module \wedge^2 M refers to the second in the list of definitions above.

 

[fresh_42]

If the characteristic is $2$, then you normally adjust the definition accordingly. E.g. instead of demanding anti-commutativity of a Lie algebra, $[X,Y]=-[Y,X]$ we demand $[X,X]=0$ instead. This is the more general case which covers all characteristics, since $0=[X+Y,X+Y]=[X,Y]+[Y,X]$ gets the usual definition from the general one.