Symmetric and Quasi-Symmetric Functions associated to Polymatroids

TL;DR

This paper introduces symmetric and quasi-symmetric functions associated with subspace arrangements and polymatroids, encoding algebraic invariants and generalizing the Tutte polynomial and matroid invariants.

Contribution

It defines new symmetric and quasi-symmetric functions for polymatroids that unify and extend existing invariants like the Tutte polynomial and matroid base polytope functions.

Findings

G[X] specializes to P[X], H[X], T[X], and F[X]

G[X] is quasi-symmetric and valuative for polymatroid decompositions

H[X] encodes Hilbert series and projective resolutions

Abstract

To every subspace arrangement X we will associate symmetric functions P[X] and H[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T[X]. Billera, Jia and Reiner recently introduced a quasi-symmetric function F[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function G[X] for polymatroids which has this property as well. Moreover, G[X] specializes to P[X], H[X], T[X] and F[X].