Koszul duality of operads and homology of partition posets

TL;DR

This paper explores the connection between Koszul duality of operads and the homology of partition posets, extending existing theories to positive characteristic fields and providing more conceptual proofs.

Contribution

It extends Koszul duality of operads to fields of positive characteristic and offers new conceptual proofs of related theorems.

Findings

Homology modules of partition posets relate to Lie representations.

Extended Koszul duality to operads over rings and positive characteristic fields.

Provided more conceptual proofs of Ginzburg and Kapranov's theorems.

Abstract

We consider partitions of a set with $r$ elements ordered by refinement. We consider the simplicial complex $\bar{K}(r)$ formed by chains of partitions which starts at the smallest element and ends at the largest element of the partition poset. A classical theorem asserts that $\bar{K}(r)$ is equivalent to a wedge of $r-1$-dimensional spheres. In addition, the poset of partitions is equipped with a natural action of the symmetric group in $r$ letters. Consequently, the associated homology modules are representations of the symmetric groups. One observes that the $r-1$th homology modules of $\bar{K}(r)$, where $r = 1,2,...$, are dual to the Lie representation of the symmetric groups. In this article, we would like to point out that this theorem occurs a by-product of the theory of \emph{Koszul operads}. For that purpose, we improve results of V. Ginzburg and M. Kapranov in several directions. More particularly, we extend the Koszul duality of operads to operads defined over a field of positive characteristic (or over a ring). In addition, we obtain more conceptual proofs of theorems of V. Ginzburg and M. Kapranov.