Symmetric and quasi-symmetric functions associated to polymatroids

Abstract

To every subspace arrangement X we will associate symmetric functions ℘[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 \({\mathcal{T}}[\mathbf{X}]\) . 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 \({\mathcal{G}}[\mathbf{X}]\) for polymatroids which has this property as well. Moreover, \({\mathcal{G}}[\mathbf{X}]\) specializes to ℘[X], ℋ[X], \({\mathcal{T}}[\mathbf{X}]\) and ℱ[X].