The Cones associated to some Transversal Polymatroids

TL;DR

This paper characterizes the facets cone of a specific transversal polymatroid, proves the associated base ring is Gorenstein, and describes related Gorenstein polymatroids in dimensions 3 and 4.

Contribution

It provides a detailed description of the facets cone for a particular class of transversal polymatroids and identifies conditions for their base rings to be Gorenstein.

Findings

The facets cone associated to the given transversal polymatroid is explicitly described.

The base ring of this polymatroid is Gorenstein, as shown using the Danilov-Stanley theorem.

Gorenstein transversal polymatroids in dimensions 3 and 4 are characterized.

Abstract

In this paper we describe the facets cone associated to transversal polymatroid presented by $\mathcal{A} = \{\{1,2\},\{2,3\},...,\{n-1,n\},\{n,1\}\}.$ Using the Danilov-Stanley theorem to characterize the canonicale module, we deduce that the base ring associated to this polymatroid is Gorenstein ring. Also, starting from this polymatroid we describe the transversal polymatroids with Gorenstein base ring in dimension 3 and with the help $\it Normaliz$ in dimension 4.