The exterior algebra of a vector space V seems to appear all over the place, such as in

- the definition of the cross product and determinant,
- the description of the Grassmannian as a variety,
- the description of irreducible representations of GL(V),
- the definition of differential forms in differential geometry,
- the description of fermions in supersymmetry.

What unifying principle lies behind these appearances of the exterior algebra? (I should mention that what I'm really interested in here is the geometric meaning of the Gessel-Viennot lemma and, by association, of the principle of inclusion-exclusion.)

This post imported from StackExchange MathOverflow at 2014-12-07 12:37 (UTC), posted by SE-user Qiaochu Yuan