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.
Highlights
Chromatic polynomials of graphs, introduced by Birkhoff and Lewis (1946) are wonderful polynomials
The primary goal of this paper is to study triune quasisymmetric functions which are Ehrhart functions, specialize to Hilbert functions, and come from combinatorial Hopf algebras
We show how principal specialization is a morphism of Hopf algebras to the ring of ‘Gaussian polynomial functions’, and that the corresponding Ehrhart ’Gaussian polynomial’ is a Hilbert function of (ΓK,h, ∆h) with respect to a certain bigrading
Summary
Chromatic polynomials of graphs, introduced by Birkhoff and Lewis (1946) are wonderful polynomials. The primary goal of this paper is to study triune quasisymmetric functions which are Ehrhart functions, specialize to Hilbert functions, and come from combinatorial Hopf algebras. Given any combinatorial Hopf monoid H with a Hopf submonoid K, there is a natural quasisymmetric function ΨK(h) associated to every element h ∈ H This invariant is a special case of the work of Aguiar et al (2006). This is motivated by the lecture notes of Grinberg and Reiner (2015), which emphasize principal specialization at q = 1. We discuss the notion of Gaussian polynomial function, which are linear combinations of polynomials in q with q-binomial coefficients
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