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Monomial and toric ideals associated to Ferrers graphs


Authors: Alberto Corso and Uwe Nagel
Journal: Trans. Amer. Math. Soc. 361 (2009), 1371-1395
MSC (2000): Primary 05A15, 13D02, 13D40, 14M25; Secondary 05C75, 13C40, 13H10, 14M12, 52B05
DOI: https://doi.org/10.1090/S0002-9947-08-04636-9
Published electronically: October 17, 2008
MathSciNet review: 2457403
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Abstract | References | Similar Articles | Additional Information

Abstract: Each partition $ \lambda = (\lambda_1, \lambda_2, \ldots, \lambda_n)$ determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a $ 2$-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulæ for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.


References [Enhancements On Off] (What's this?)

  • 1. S.S. Abhyankar, Enumerative combinatorics of Young tableaux, Marcel Dekker, New York, Basel, 1988. MR 926272 (89e:05011)
  • 2. F. Brenti, G. Royle and D. Wagner, Location of zeros of chromatic and related polynomials of graphs, Canad. J. Math. 46 (1994), 55-80. MR 1260339 (94k:05077)
  • 3. D. Bayer and B. Sturmfels, Cellular resolutions of monomial modules, J. Reine Angew. Math. 502 (1998), 123-140. MR 1647559 (99g:13018)
  • 4. D. Bayer, I. Peeva and B. Sturmfels, Monomial resolutions, Math. Res. Lett. 5 (1998), 31-46. MR 1618363 (99c:13029)
  • 5. R. Biagioli, S. Faridi and M. Rosas, Resolutions of De Concini-Procesi ideals of hooks, Communications in Algebra 35 (2007), 3875-3891. MR 2371263
  • 6. W. Bruns and A. Guerrieri, The Dedekind-Mertens formula and determinantal rings, Proc. Amer. Math. Soc. 127 (1999), 657-663. MR 1468185 (99f:13013)
  • 7. W. Bruns and J. Herzog, On the computation of $ a$-invariants, Manuscripta Math. 77 (1992), 201-213. MR 1188581 (93k:13032)
  • 8. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
  • 9. F. Butler, Rook theory and cycle-counting permutation statistics, Adv. in Appl. Math. 33 (2004), 655-675. MR 2095860 (2005j:05002)
  • 10. A. Conca, Ladder determinantal rings, J. Pure Appl. Algebra 98 (1995), 119-134. MR 1319965 (96a:13013)
  • 11. A. Conca, Straightening law and powers of determinantal ideals of Hankel matrices, Adv. Math. 138 (1998), 263-292. MR 1645574 (99i:13020)
  • 12. A. Conca and J. Herzog, On the Hilbert function of determinantal rings and their canonical module, Proc. Amer. Math. Soc. 112 (1994), 677-681. MR 1213858 (95a:13016)
  • 13. A. Corso and U. Nagel, Specializations of Ferrers ideals, to appear in J. Algebraic Combin.
  • 14. A. Corso, W.V. Vasconcelos and R.H. Villarreal, Generic Gaussian ideals, J. Pure Appl. Algebra 125 (1998), 117-127. MR 1600012 (98m:13014)
  • 15. M. Develin, Rook poset equivalence of Ferrers boards, Order 23 (2006), 179-195. MR 2308905
  • 16. K. Ding, Rook placements and cellular decomposition of partition varieties, Discrete Mathematics 170 (1997), 107-151. MR 1452940 (98i:05166)
  • 17. J.A. Eagon and D.G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. Ser. A 269 (1962), 188-204. MR 0142592 (26:161)
  • 18. R. Ehrenborg and S. van Willigenburg, Enumerative properties of Ferrers graphs, Discrete Comput. Geom. 32 (2004), 481-492. MR 2096744 (2005j:05076)
  • 19. D. Eisenbud, M. Green, K. Hulek and S. Popescu, Restricting linear syzygies: Algebra and geometry, Compositio Math. 141 (2005), 1460-1478. MR 2188445 (2006m:14072)
  • 20. D. Eisenbud, M. Green, K. Hulek and S. Popescu, Small schemes and varieties of minimal degree, Amer. J. Math. 128 (2006), 1363-1389. MR 2275024 (2007j:14078)
  • 21. R. Fröberg, On Stanley-Reisner rings, in Topics in algebra, Part 2 (Warsaw, 1988), pp. 57-70, Banach Center Publ. 26, PWN, Warsaw, 1990. MR 1171260 (93f:13009)
  • 22. I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, Adv. Math. 58 (1985), 300-321. MR 815360 (87e:05008)
  • 23. J. Goldman, J.T. Joichi and D. White, Rook Theory I. Rook equivalence of Ferrers boards, Proc. Amer. Math. Soc. 52 (1975), 485-492. MR 0429578 (55:2590)
  • 24. E. Gorla, Mixed ladder determinantal varieties from two-sided ladders, J. Pure Appl. Algebra 211, (2007) 433-444. MR 2340461
  • 25. H.W. Gould, Combinatorial identities: A standardized set of tables listing 500 binomial coefficient summations (rev. ed.), Morgantown, West Virginia, 1972. MR 0354401 (50:6879)
  • 26. S.R. Ghorpade, Hilbert functions of ladder determinantal varieties, Discrete Math. 246 (2002), 131-175. MR 1887484 (2003e:13026)
  • 27. D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, in Combinatoire Enumerative (Montreal, 1985), pp. 112-125, Lect. Notes Math. 1234, 1986. MR 927762 (89d:05005)
  • 28. H.T. Hà and A. Van Tuyl, Splittable ideals and the resolutions of monomial ideals, J. Algebra 309 (2007), 405-425. MR 2301246 (2008a:13016)
  • 29. J. Haglund, Rook theory and hypergeometric series, Adv. in Appl. Math. 17 (1996), 408-459. MR 1422066 (98k:33010)
  • 30. J. Herzog and T. Hibi, Distributive lattices, bipartite graphs and Alexander duality, J. Algebraic Combin. 22 (2005), 289-302. MR 2181367 (2006h:06004)
  • 31. J. Herzog, T. Hibi and X. Zheng, Monomial ideals whose powers have a linear resolution, Math. Scand. 95 (2004), 23-32. MR 2091479 (2005f:13012)
  • 32. J. Herzog and N.V. Trung, Gröbner bases and multiplicity of determinantal and Pfaffian ideals, Adv. Math. 96 (1992), 1-37. MR 1185786 (94a:13012)
  • 33. J. Kleppe, J. Migliore, R.M. Miró-Roig, U. Nagel and C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154 (2001), no. 732. MR 1848976 (2002e:14083)
  • 34. C. Krattenthaler, Non-crossing two-rowed arrays and summations for Schur functions, preprint 1992.
  • 35. C. Krattenthaler and S.G. Mohanty, On lattice path counting by major index and descents, Europ. J. Combin. 14 (1993), 43-51. MR 1197475 (94e:05019)
  • 36. C. Krattenthaler and M. Prohaska, A remarkable formula for counting non-intersecting lattice paths in a ladder with respect to turns, Trans. Amer. Math. Soc. 351 (1999), 1015-1042. MR 1407495 (99e:05009)
  • 37. C. Krattenthaler and M. Rubey, A determinantal formula for the Hilbert series of one-sided ladder determinantal rings, in Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), pp. 525-551, Springer, Berlin, 2004. MR 2037108 (2005a:13026)
  • 38. D.M. Kulkarni, Counting of paths and coefficients of the Hilbert polynomial of a determinantal ideal, Discrete Math. 154 (1996), 141-151. MR 1395454 (97f:13018)
  • 39. J. Migliore, Introduction to liaison theory and deficiency modules, Progress in Mathematics 165, Birkhäuser, 1998. MR 1712469 (2000g:14058)
  • 40. J. Migliore, U. Nagel and T. Römer, Extensions of the multiplicity conjecture, to appear in Trans. Amer. Math. Soc.
  • 41. E. Miller and B. Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer-Verlag, New York, 2005. MR 2110098 (2006d:13001)
  • 42. A. Mitchell, The inverse rook problem on Ferrers boards, preprint 2004.
  • 43. H. Narasimhan, The irreducibility of ladder determinantal varieties, J. Algebra 102 (1986), 162-185. MR 853237 (87j:14081)
  • 44. C. Polini, B. Ulrich and M. Vitulli, The core of zero-dimensional monomial ideals, Adv. Math. 211 (2007), 72-93. MR 2313528 (2008b:13033)
  • 45. M. Rubey, The $ h$-vector of a ladder determinantal ring cogenerated by $ 2\times 2$ minors is log-concave, J. Algebra 292 (2005), 303-323. MR 2172157 (2007a:05147)
  • 46. A. Simis, W.V. Vasconcelos and R. Villarreal, On the ideal theory of graphs, J. Algebra 167 (1994), 389-416. MR 1283294 (95e:13002)
  • 47. R.P. Stanley, Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999. MR 1676282 (2000k:05026)
  • 48. R.P. Stanley, Catalan addendum (URL: www-math.mit.edu/~rstan/ec/catadd.pdf).
  • 49. C. Valencia and R. Villarreal, Canonical modules of certain edge subrings, European J. Combin. 24 (2003), 471-487. MR 1983672 (2004c:13033)
  • 50. A. Varvak, Rook numbers and the normal ordering problem, J. Combin. Theory Ser. A 112 (2005), 292-307. MR 2177488 (2006h:05011)
  • 51. R. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics 238, Marcel Dekker, Inc., New York, 2001. MR 1800904 (2002c:13001)
  • 52. H.-J. Wang, A determinantal formula for the Hilbert series of determinantal rings of one-sided ladder, J. Algebra 265 (2003), 79-99. MR 1984900 (2004d:13016)
  • 53. H.-J. Wang, Counting of paths and the multiplicity of determinantal rings, preprint 2002.
  • 54. H.-J. Wang, A conjecture of Herzog and Conca on counting of paths, preprint 2002.

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Additional Information

Alberto Corso
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: corso@ms.uky.edu

Uwe Nagel
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: uwenagel@ms.uky.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04636-9
Received by editor(s): January 22, 2007
Published electronically: October 17, 2008
Additional Notes: The second author gratefully acknowledges partial support from the NSA under grant H98230-07-1-0065
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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