Standard multigraded Hibi rings and Cartwright-Sturmfels ideals

TL;DR

This paper studies standard multigradings on Hibi rings, showing they are induced by poset chains, and characterizes when their ideals are Cartwright-Sturmfels, with calculations of Hilbert series and multidegree polynomials.

Contribution

It introduces a classification of multigradings on Hibi rings and characterizes Cartwright-Sturmfels ideals within this context, extending the understanding of their algebraic properties.

Findings

Multigradings on Hibi rings are induced by poset chains.

The multigraded Hilbert series of Hibi rings are computed.

Hibi ideals that are Cartwright-Sturmfels are characterized.

Abstract

In this paper, we introduce standard multigradings on Hibi rings, which are algebras arising from posets. We show that any standard multigrading on a Hibi ring that makes its defining ideal (called the Hibi ideal) homogeneous is induced by a chain of the underlying poset. After that, we calculate the multigraded Hilbert series of Hibi rings by generalizing the theory of $P$-partition and we compute the multidegree polynomials of Hibi rings. Furthermore, we characterize Hibi ideals that are Cartwright-Sturmfels ideals.