Letterplace and co-letterplace ideals of posets

Abstract

To a natural number $n$, a finite partially ordered set $P$ and a poset ideal ${\mathcal J}$ in the poset $Hom(P,[n])$ of isotonian maps from $P$ to the chain on $n$ elements, we associate two monomial ideals, the letterplace ideal $L(n,P;{\mathcal J})$ and the co-letterplace ideal $L(P,n;{\mathcal J})$. These ideals give a unified understanding of a number of ideals studied in monomial ideal theory in recent years. By cutting down these ideals by regular sequences of variable differences we obtain: multichain ideals and generalized Hibi type ideals, initial ideals of determinantal ideals, strongly stable ideals, $d$-partite $d$-uniform ideals, Ferrers ideals, edge ideals of cointerval $d$-hypergraphs, and uniform face ideals.Comment: 24 pages, updated introduction and references, and some minor improvement

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

“…We say it has left strict chain fibers if φ −1 (r) ⊆ S is a chain when considered in P op × N and all its elements have distinct first coordinates. Again we prove this in Section 7, and it is a generalization of Theorem 5.12 in [6]. Its proof also follows rather closely the proof in [6].…”

“…Let [n] = {1 < 2 < · · · < n}. In [6] the setting was a poset ideal J ⊆ Hom(P, [n]) and we defined letterplace and co-letterplace ideals L(n, P ; J ) and L(P, n; J ) in the polynomial ring k[x P × [n] ]. This corresponds to finite poset ideals J in the definition above, see Section 4.…”

“…These give the principal letterplace ideals L(α, P ) and co-letterplace ideals L(P, α). The ideals L(n, P ) and L(P, n) of [6] are the special cases when α is the constant map sending each p to n. When P is the antichain L(α, P ) is a complete intersection, and these ideals may therefore be considered as generalizations of complete intersections. Determinantal ideals.…”

“…A particular case of principal letterplace ideals is when P is the chain [m]. In [6] we showed that L(n, [m]) is the initial ideal of the ideal of maximal minors of an (m + n − 1) × n-matrix. Here we show that L(α, [m]) is also an initial ideal of a class of determinantal ideals.…”

Alexander duality for the alternative polarizations of strongly stable ideals

“…The next two results are implicitly contained in Fløystad [5]. However they are stated in the context of the preceding papers [6,2], where the words "letterplace ideal" and "coletterplace ideals" are used in the narrow sense (see Remark 2.8 below). Proof.…”