Standard multigraded Hibi rings and Cartwright-Sturmfels ideals

TL;DR

This work develops a canonical framework for multigradings on Hibi rings $S_L/I_L$ by tying any homogeneous multigrading to a chain of the underlying poset via a map $f_C$. It derives the multigraded Hilbert series using generalized $P$-partitions and computes multidegree polynomials through maximal chains of the distributive lattice $L(P)$, linking combinatorics of posets to multigraded invariants. A fundamental contribution is the complete characterization of when Hibi ideals are Cartwright–Sturmfels under these gradings: precisely when $P\setminus C$ is a chain, with a dichotomy argument based on $2$-minors and elimination. These results illuminate the interplay between poset structure, multigradings, and algebraic properties like initial ideals, enabling explicit computations across varied multigradings.

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.

Figures (7)

• Figure 1: The poset $P$
• Figure 2: The distributive lattice $L(P)$
• Figure 3: Multigrading by $C_1$
• Figure 4: Multigrading by $C_2$
• Figure 5: $P$

Theorems & Definitions (25)

• Proposition 2.1
• proof
• Example 2.2
• Theorem 2.3
• proof
• Lemma 3.1: StanleyCombinatorics
• Theorem 3.2
• proof
• Corollary 3.3
• Example 3.4