Why is the exterior algebra so ubiquitous?

Question

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

Answer 1

A point which is perhaps worthwile of explicite mention is that the symmetric group (where we consider only the subgroup of all elements with finite support if $E$ is infinite) of a set $E$ has two one-dimensional representations. The trivial representation is of course related to the symmetric algebra and the signature representation corresponds to the exterior algebra.

This post imported from StackExchange MathOverflow at 2014-12-07 12:38 (UTC), posted by SE-user Roland Bacher

Answer 2

It is simply because geometry ( in the abstract sense) is the foundation of physics and exterior algebra is just a subalgebra of geometric algebra which unifies all geometric concepts across all areas of mathematics.