TITLE: Using Exterior Algebras in combinatorics
QUESTION [10 upvotes]: 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?

REPLY [2 votes]: 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 non-commutative 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, Cauchy-Binet formulae, and Capelli-type 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. 93-102, 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 so-called Jacobi, Sylvester theorems
for some matrices with non-necessary 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