The colored symmetric and exterior algebras (extended abstract)

R. S. González D'León

Proceedings of FPSAC 2016.

Abstract:

In this extended abstract we present 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 describe 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 show that the two colored Koszul dual algebras are Koszul in the sense of Priddy et al.