The colored symmetric and exterior algebras

TL;DR

This paper explores colored generalizations of symmetric and exterior algebras, revealing their representation theory, cohomological properties, and Koszul duality using poset topology and algebraic techniques.

Contribution

It introduces the weighted boolean algebra poset to analyze the symmetric group actions and provides explicit formulas for the representations of the colored exterior algebra.

Findings

Colored exterior algebra's multilinear components are isomorphic to top cohomology modules.

The Frobenius series of the colored exterior algebra is explicitly computed.

Both colored symmetric and exterior algebras are proven to be Koszul.

Abstract

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.