Algebraic properties of classes of path ideals

TL;DR

This paper explores algebraic properties of path ideals derived from specific posets like trees and cycles, linking their Cohen-Macaulayness to the poset structure and analyzing invariants of related monomial ideals in algebraic statistics.

Contribution

It characterizes when path ideals are sequentially Cohen-Macaulay based on the underlying poset structure and computes key invariants for Luce-decomposable monomial ideals.

Findings

Path ideals of certain posets are sequentially Cohen-Macaulay under specific conditions.

Explicit formulas for Krull dimension, projective dimension, regularity, and Betti numbers are provided for Luce-decomposable ideals.

The study connects combinatorial poset properties with algebraic invariants of associated monomial ideals.

Abstract

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen-Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise from the Luce-decomposable model in algebraic statistics, can be viewed as path ideals of certain posets. We study invariants of these so-called \emph{Luce-decomposable} monomial ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their regularity and their Betti numbers.