Conic divisorial ideals of toric rings and applications to Hibi rings and stable set rings

TL;DR

This paper investigates conic divisorial ideals in toric rings, especially Hibi and stable set rings, providing new methods for their description and applications to resolutions and symmetry properties.

Contribution

It introduces a novel approach to determine conic divisorial ideals and applies it to characterize symmetry and construct NCCRs for specific stable set rings.

Findings

Described conic divisorial ideals for Hibi and stable set rings.

Characterized when these rings are quasi-symmetric or weakly-symmetric.

Constructed a non-commutative crepant resolution for certain stable set rings.

Abstract

In the present paper, we study conic divisorial ideals of toric rings. We provide an idea to determine them and we give a description of the conic divisorial ideals of Hibi rings and stable set rings of perfect graphs by using this idea. We also characterize when Hibi rings or stable set rings are quasi-symmetric or weakly-symmetric. Moreover, by using the description of the conic divisorial ideals, we construct a non-commutative crepant resolution (NCCR) of a special family of stable set rings.