The type of the base ring associated to a transversal polymatroid

TL;DR

This paper characterizes the facets of the polyhedral cone linked to a transversal polymatroid's base ring, enabling computation of algebraic invariants like the canonical module and $a$-invariant, through computational experiments.

Contribution

It provides a detailed description of the cone's facets for certain transversal polymatroids, advancing understanding of their algebraic properties.

Findings

Facets of the cone are explicitly determined.

Canonical module expressed via the cone's relative interior.

Computed $a$-invariant for these base rings.

Abstract

In this paper we determine the facets of the polyhedral cone generated by the exponent set of the monomials defining the base ring associated to some transversal polymatroid. We need the description of these facets to find the canonical module of the base ring which is expressed in terms of the relative interior of the cone. This would allow us to compute the $a$-invariant of those base rings. The results presented were discovered by extensive computer algebra experiments performed with {\it{Normaliz}} \cite{BK}.