Cohomology of complete unordered flag manifolds

TL;DR

This paper computes the cohomology of quotients of complete flag manifolds by symmetric group actions, revealing homological stability and providing explicit descriptions of stable and unstable cohomology.

Contribution

It introduces a method to compute the cohomology of these quotient spaces and describes their stable cohomology rings explicitly.

Findings

Spaces exhibit homological stability

Stable cohomology rings are explicitly described

An algorithm for unstable cohomology computation is provided

Abstract

We consider quotients of complete flag manifolds in Cn and Rn by an action of the symmetric group on n objects. We compute their cohomology with field coefficients of any characteristic. Specifically, we show that these topological spaces exhibit homological stability and we provide a closed-form description of their stable cohomology rings. We also describe a simple algorithmic procedure to determine their unstable cohomology additively.