Quasi-symmetric functions and the Chow ring of the stack of expanded pairs

TL;DR

This paper reveals that the Hopf algebra of quasi-symmetric functions naturally appears as the Chow ring of a specific algebraic stack, connecting combinatorics with algebraic geometry.

Contribution

It establishes a new link between quasi-symmetric functions and the Chow ring of the stack of expanded pairs, providing a combinatorial perspective and explicit calculations.

Findings

Chow ring of the stack of expanded pairs is isomorphic to the Hopf algebra of quasi-symmetric functions.

A gluing map induces the comultiplication in the Hopf algebra.

Chow rings of certain stacks of semistable curves are explicitly computed.

Abstract

We show that the Hopf algebra of quasi-symmetric functions arises naturally as the integral Chow ring of the algebraic stack of expanded pairs originally described by J. Li, using a more combinatorial description in terms of configurations of line bundles. In particular, we exhibit a gluing map which gives rise to the comultiplication. We then apply the result to calculate the Chow rings of certain stacks of semistable curves.