As addressed in this past question, there are many applications of linear algebra to combinatorics. What about examples of applications of exterior algebras? Part 4 here is one such example. What are the heuristics for when to apply them and the intuition for how to do so?

This is a bit of cheating (since requires to know a bit of homological algebra) but is too nice to not be mentioned:



Gil Kalai's theory of algebraic shifting (see for instance http://www.ma.huji.ac.il/~kalai/japan.ps) gives many applications of exterior algebra to combinatorics. 


The exterior algebra arose naturally in a problem about enumerating matchings on a planar graph (domino/lozenge tilings). In some situations, those correspond to families of nonintersecting lattice paths between a set of sources $S$ and a set of sinks $T$. The LindstromGesselViennot lemma says that the number of these families equals the determinant of a matrix $M$ whose $i,j$ term is the number of paths between the $i$th element of $S$ and the $j$th element of $T$. Suppose you want to count the domino tilings of a modified region where only some subset of $S$ is present and only some subset of the same size of $T$ is present. This happens if you have a large region you cut into pieces along $S$ and $T$, and want to enumerate tilings of the larger region by the transfer matrix method. If you index a matrix by subsets of $S$ and subsets of $T$, and put the number of domino tilings of the region in that entry, the result is the action of $M$ on the exterior algebra. 


The socalled MacMahon master theorem (also called Wronski identity) and its generalizations (q,super, ...) e.g. Zeilberger et. al. and e.g. Foata et. al. can be The main example of Koszul duality is duality between exterior algebra and polynomial algebra, it is enough to obtain the classical MacMahon identity as well as its generalization to certain matrices with noncommuting entries (Manin matrices). There are further qsuper analogs  everything is the same just need to consider appropriate versions of the exterior algebra. Let me sketch the idea of application which is quite simple, it is related to Vladimir's answer above. One knows that Euler characteristic can be calculated from complex itself and from (co)homology of the complex  we get the same result. This actually can be generalized to the trace of arbitrary operator acting on the complex, if it commutes with differential. So the proof of the MacMahon formula exploits this idea for the Koszul complex, which cohomology consits of $C$. So the trace taken over cohomology is $1$, while the trace calculated via the complex itself will give $det(1A)(\sum_k Tr S^k A)$. The advantage of this proof that it can be generelized to matrices with noncommuting entries  Manin matrices, quantum matrices, super matrices, etc... While the standard proof via the diagonalization will not work in such situations. See also An algebraic extension of the MacMahon Master Theorem Pavel Etingof, Igor Pak This also has some applications in representation theory and quantum integrable systems The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for \hat gl(mn) A. I. Molev, E. Ragoucy; 


Probably the main thing about the exterior algebra is that it is intimately related to the determinats of matrices, and it can be effeciently used to obtain certain statements which includes matrix determinants like det(AB)=det(A)det(B), Cramer's formula for A^{1}, Plucker relations, Jacobi ratio theorem, etc. (Example on MO). On the other hand there are cerain amount of papers from combintatorial community (names include G.C. Rota, D. Zeilberger, D. Foata) to prove similar relations by combinatorial means. Hope this might be considered as an application or at least as a relation. Before giving some concrete examples let me mention that both techniques (combinatorial and exterior algebra) allows to handle matrices with noncommutative elements, which is of certain interest in representation theory and quantum integrable systems. Example 1. Capelli identity det(A)det(B) = det(AB+ correction) There is a paper Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory Dominique Foata, Doron Zeilberger And some powerful generalization: Noncommutative determinants, CauchyBinet formulae, and Capellitype identities Sergio Caracciolo, Andrea Sportiello, Alan D. Sokal Both combinatorial in nature. Grassman algebra proof and genealization were given in Algebraic properties of Manin matrices . Let me mention that G.C. Rota was also interested in Capelli identies (G.C. Rota, Combinatirial theory and invariant theory, NSF advanced theory in combinatorial theory, Bowdain College, Maine, Notes by L. Guibas, 1971). Which influenced S. G. Williamson, Symmetry operators, polarizations, and a generalized Capelli identity, Linear Multilinear Algebra , vol. 10, no. 2, pp. 93102, 1981 DOI: 10.1080/03081088108817399 Remarkable paper which is undesrvedly forgotten, results partly rediscovered by A. Okounkov and M. Nazarov decades later. Example 2. Matjaž Konvalinka (who was a student of Igor Pak in MIT at that time) obtained combinatorially the proofs the socalled Jacobi, Sylvester theorems for some matrices with nonnecessary commutative entries (Manin matrices). (These papers are motivated by other Zeilberger's&K paper  I already discussed this in another answer to this question). Exterior algebra proofs can be found again here: Algebraic properties of Manin matrices 


There's something called the "infinite wedge" you may wish to look into. There are a number of deep applications to counting random partitions and branched covers of surfaces. 

