The quasisymmetric flag variety: a toric complex on noncrossing partitions

TL;DR

This paper introduces a new geometric structure called the quasisymmetric flag variety, linking noncrossing partitions with quasisymmetric functions through a toric complex in the flag variety.

Contribution

It develops the geometric theory of equivariant quasisymmetry and constructs the quasisymmetric flag variety, connecting noncrossing partitions with the cohomology ring of quasisymmetric coinvariants.

Findings

The quasisymmetric flag variety is a toric complex with fixed points corresponding to noncrossing partitions.

The cohomology ring of this variety is isomorphic to the ring of quasisymmetric coinvariants.

This framework provides a geometric perspective on quasisymmetry and noncrossing partitions.

Abstract

We develop the geometric theory of equivariant quasisymmetry via a new ``quasisymmetric flag variety''. This is a toric complex in the flag variety whose fixed point set is the set of noncrossing partitions, and whose cohomology ring is the ring of quasisymmetric coinvariants.