Poset ideals of P-partitions and generalized letterplace and determinantal ideals

TL;DR

This paper explores monomial ideals associated with poset ideals of P-partitions, establishing dualities, and introducing new classes of determinantal ideals that generalize maximal minors.

Contribution

It introduces the concept of P-stable monomial ideals, establishes dualities for strongly stable ideals when P is a chain, and constructs new determinantal ideals generalizing maximal minors.

Findings

Defined P-stable monomial ideals.

Established duality for strongly stable ideals in chain posets.

Constructed new determinantal ideals generalizing maximal minors.

Abstract

For any finite poset $P$ we have the poset of isotone maps $\text{Hom}(P,\mathbb{N})$, also called $P^{op}$-partitions. To any poset ideal ${\mathcal J}$ in $\text{Hom}(P,\mathbb{N})$, finite or infinite, we associate monomial ideals: the letterplace ideal $L({\mathcal J},P)$ and the Alexander dual co-letterplace ideal $L(P,{\mathcal J})$, and study them. We derive a class of monomial ideals in $k[x_p, p \in P]$ called $P$-stable. When $P$ is a chain we establish a duality on strongly stable ideals. We study the case when ${\mathcal J}$ is a principal poset ideal. When $P$ is a chain we construct a new class of determinantal ideals which generalizes ideals of {\it maximal} minors and whose initial ideals are letterplace ideals of prinicpal poset ideals.