Advertisement

Acta Mathematica Vietnamica

, Volume 44, Issue 1, pp 213–241 | Cite as

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

  • Gunnar FløystadEmail author
Article
  • 32 Downloads

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.

Keywords

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

Mathematics Subject Classification (2010)

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

References

  1. 1.
    Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes in Mathematics, vol. 1327. Springer, Berlin (1988)Google Scholar
  2. 2.
    D’Alì, A., Fløystad, G., Nematbakhsh, A.: Resolutions of co-letterplace ideals and generalizations of Bier spheres. To appear in Trans. Am. Math. Soc. arXiv:1601.02793 (2016)
  3. 3.
    Ene, V.: Syzygies of Hibi rings. Acta Math. Vietnam. 40(3), 403–446 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ene, V., Herzog, J., Mohammadi, F.: Monomial ideals and toric rings of Hibi type arising from a finite poset. Eur. J. Comb. 32(3), 404–421 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Féray, V., Reiner, V.: P-partitions revisited. J. Commut. Algebra 4(1), 101–152 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fløystad, G., Møller Greve, B., Herzog, J.: Letterplace and co-letterplace ideals of posets. J. Pure Appl. Algebra 221(5), 1218–1241 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fløystad, G., Nematbakhsh, A.: Rigid ideals by deforming quadratic letterplace ideals. To appear in J. Algebra. arXiv:1606.07417 (2016)
  8. 8.
    Francisco, C.A., Mermin, J., Schweig, J.: Generalizing the Borel property. J. Lond. Math. Soc. (2) 87(3), 724–740 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garsia, A.M.: Combinatorial methods in the theory of Cohen-Macauly rings. Adv. Math. 38(3), 229–266 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Herzog, J., Popescu, D.: Finite filtrations of modules and shellabe multicomplexes. Manuscr. Math. 121(3), 385–410 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Herzog, J., Trung, N.V.: Gröbner bases and multiplicity of determinantal and Pfaffian ideals. Adv. Math. 96(1), 1–37 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Stanley, R.P.: Ordered Structures and Partitions, vol. 119. American Mathematical Society, Providence (1972)Google Scholar
  13. 13.
    Stanley, R.P.: Enumerative Combinatorics. Cambridge Studies in Advanced Mathematics 1. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  14. 14.
    Sturmfels, B.: Gröbner bases and Stanley decompositions of determinantal rings. Math. Z. 205(1), 137–144 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Matematisk InstituttUniversitetet i BergenBergenNorway

Personalised recommendations