  We study colored generalizations of the symmetric algebra and its Koszul
dual, the exterior algebra. The symmetric group $\mathfrak{S}_n$ acts on the
multilinear components of these algebras. While $\mathfrak{S}_n$ acts trivially
on the multilinear components of the colored symmetric algebra, we use poset
topology techniques to understand the representation on its Koszul dual. We
introduce an $\mathfrak{S}_n$-poset of weighted subsets that we call the
weighted boolean algebra and we prove that the multilinear components of the
colored exterior algebra are $\mathfrak{S}_n$-isomorphic to the top cohomology
modules of its maximal intervals. We use a technique of Sundaram to compute
group representations on Cohen-macaulay posets to give a generating formula for
the Frobenius series of the colored exterior algebra. We exploit that formula
to find an explicit expression for the expansion of the corresponding
representations in terms of irreducible $\mathfrak{S}_n$-representations. We
show that the two colored Koszul dual algebras are Koszul in the sense of
Priddy.
