TITLE: Why is the exterior algebra so ubiquitous?
QUESTION [59 upvotes]: 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.)

REPLY [2 votes]: I think that what unifies some of the different examples of when the exterior algebra occurs is that it is the structure that transforms the action of a commutative ring on a module (or, more concretely, the action of several commuting linear operators on a vector space) into a chain complex. The ubiquitous structure is really commutative rings and modules over commutative rings.  The exterior algebra actually encodes commutativity, in a sense. (One might then ask why is it that mathematicians love so much commutative rings, commuting operators, etc.)
When tensoring a commutative action with the graded algebra one gets the Koszul complex, and the anti-commutative nature of the graded algebra is precisely what, when coupled with the commutative action of the ring, makes possible the definition of a dirac operator $d$ with $d^2=0$.
I got these ideas after thinking about Joseph Taylor's paper, "A joint spectrum for several commuting operators", J. Funct. Anal. , 1970.