The antisymmetrization map Altπ‘˜:𝑇0 k(𝑉)β†’πΏπ‘˜alt(𝑉) is defined by

Altπ‘˜(πœ”)(𝑣1,…,π‘£π‘˜)≔1π‘˜!βˆ‘πœŽβˆˆπ‘†π‘˜sgn(𝜎)πœ”(π‘£πœŽ1,…,π‘£πœŽπ‘˜)

Note

This a is map from tensors to space of alternating maps.