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.

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Correspondence to Gunnar Fløystad.

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Fløystad, G. Poset Ideals of P-Partitions and Generalized Letterplace and Determinantal Ideals. Acta Math Vietnam 44, 213–241 (2019). https://doi.org/10.1007/s40306-018-0262-3

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Keywords

  • Poset ideals
  • Letterplace ideals
  • P-partitions
  • Strongly stable ideals
  • Determinantal ideals

Mathematics Subject Classification (2010)

  • Primary 13F55, 05E40
  • Secondary 13C40, 14M12