Poset Ideals of P-Partitions and Generalized Letterplace and Determinantal Ideals

Abstract

For any finite poset P, we have the poset of isotone maps $\text {Hom}(P,\mathbb {N})$, also called $P^{\text {op}}$-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 $\Bbbk [x_{p}, p \in P]$ 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 maximal minors and whose initial ideals are letterplace ideals of principal poset ideals.