Quasisymmetric Functions from Combinatorial Hopf Monoids and Ehrhart Theory

TL;DR

This paper explores the connection between quasisymmetric functions derived from combinatorial Hopf monoids, Ehrhart theory, and Hilbert functions of relative simplicial complexes, revealing new links across algebraic and geometric combinatorics.

Contribution

It demonstrates that quasisymmetric functions from Hopf monoids naturally relate to Ehrhart theory and Hilbert functions, establishing a novel interdisciplinary connection.

Findings

Quasisymmetric functions from Hopf monoids relate to Ehrhart theory.

Specializations of these functions serve as Hilbert functions for certain complexes.

Forbidden composition complexes form a Hopf monoid, linking algebra and geometry.

Abstract

We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.