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.)



Why is the exterior algebra so ubiquitous?
Just to use a buzzword that Greg didn't, the exterior algebra is the symmetric algebra of a purely odd supervector space. So, it isn't "better than a symmetric algebra," it is a symmetric algebra.


The reason this happens is that super vector spaces aren't just Z/2 graded vector spaces, they also have a slightly different tensor category structure (the flip map on the tensor product of two odd vector spaces is -1 times the usual flip map, and the usual flip map for all other pure vector spaces). If you look at all the formulas from homological algebra, for things like how to take the tensor product of two complexes, they always have a bunch of weird signs showing up; these always can be thought of as coming from the fact that you should take the tensor product on graded vector spaces inherited from super vector spaces, not the boring one.


Of course, this just raises the question of why supervector spaces show up so much. Greg had about as good an answer as I could give for that.


