On the Möbius function of a lower Eulerian Cohen-Macaulay poset
Christos A. Athanasiadis
Department of Mathematics, University of Athens, Athens, 15784 Hellas, Greece
DOI: 10.1007/s10801-011-0306-4
Abstract
A certain inequality is shown to hold for the values of the Möbius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen-Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley's nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen-Macaulay meet-semilattice and Adin's nonnegativity conjecture for the cubical h-vector of a Cohen-Macaulay cubical complex.
Pages: 373–388
Keywords: Eulerian poset; Cohen-Macaulay poset; Möbius function; cubical $h$-vector; toric $h$-vector; Buchsbaum complex
Full Text: PDF
References
1. Adin, R.M.: A new cubical h-vector. Discrete Math. 157, 3-14 (1996)
2. Athanasiadis, C.A., Welker, V.: Buchsbaum* complexes. To appear in Math. Z.
3. Babson, E.K., Billera, L.J., Chan, C.S.: Neighborly cubical spheres and a cubical lower bound conjecture. Isr. J. Math. 102, 297-315 (1997)
4. Björner, A.: Topological methods. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, pp. 1819-1872. North Holland, Amsterdam (1995)
5. Browder, J., Klee, S.: Lower bounds for Buchsbaum* complexes. Eur. J. Comb. 32, 146-153 (2011)
6. Fløystad, G.: Cohen-Macaulay cell complexes. In: Algebraic and Geometric Combinatorics. Contemporary Mathematics, vol. 423, pp. 205-220. American Mathematical Society, Providence (2007)
7. Hetyei, G.: Invariants des complexes cubiques. Ann. Sci. Math. Québec 20, 35-52 (1996)
8. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Reading (1984)
9. Stanley, R.P.: The upper bound conjecture and Cohen-Macaulay rings. Stud. Appl. Math. 54, 135-142 (1975)
10. Stanley, R.P.: Cohen-Macaulay complexes. In: Aigner, M. (ed.) Higher Combinatorics, pp. 51-62. Reidel, Dordrecht/Boston (1977)
11. Stanley, R.P.: Balanced Cohen-Macaulay complexes. Trans. Am. Math. Soc. 249, 139-157 (1979)
12. Stanley, R.P.: Enumerative Combinatorics, vol.
1. Wadsworth & Brooks/Cole, Pacific Grove (1986). Second printing, Cambridge Studies in Advanced Mathematics, vol.
49. Cambridge University Press, Cambridge (1997)
13. Stanley, R.P.: Generalized h-vectors, intersection cohomology of toric varieties and related results. In: Nagata, N., Matsumura, H. (eds.) Commutative Algebra and Combinatorics. Adv. Stud. Pure Math., vol. 11, pp. 187-213. Kinokuniya/North-Holland, Tokyo/Amsterdam/New York (1987)
14. Stanley, R.P.: f -vectors and h-vectors of simplicial posets. J. Pure Appl. Algebra 71, 319-331 (1991)
15. Stanley, R.P.: Combinatorics and Commutative Algebra, 2nd edn. Birkhäuser, Basel (1996) J Algebr Comb (2012) 35:373-388
16. Stanley, R.P.: Positivity problems and conjectures in algebraic combinatorics. In: Arnold, V., Atiyah, M., Lax, P., Mazur, B. (eds.) Mathematics: Frontiers and Perspectives, pp. 295-319. Amer. Math. Society, Providence (2000)
17. Walker, J.: Canonical homeomorphisms of posets. Eur. J. Comb. 9, 97-107 (1988)
18. Yanagawa, K.: Higher Cohen-Macaulay property of squarefree modules and simplicial posets. Proc. Am. Math. Soc. 139, 3057-3066 (2011).
2. Athanasiadis, C.A., Welker, V.: Buchsbaum* complexes. To appear in Math. Z.
3. Babson, E.K., Billera, L.J., Chan, C.S.: Neighborly cubical spheres and a cubical lower bound conjecture. Isr. J. Math. 102, 297-315 (1997)
4. Björner, A.: Topological methods. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, pp. 1819-1872. North Holland, Amsterdam (1995)
5. Browder, J., Klee, S.: Lower bounds for Buchsbaum* complexes. Eur. J. Comb. 32, 146-153 (2011)
6. Fløystad, G.: Cohen-Macaulay cell complexes. In: Algebraic and Geometric Combinatorics. Contemporary Mathematics, vol. 423, pp. 205-220. American Mathematical Society, Providence (2007)
7. Hetyei, G.: Invariants des complexes cubiques. Ann. Sci. Math. Québec 20, 35-52 (1996)
8. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Reading (1984)
9. Stanley, R.P.: The upper bound conjecture and Cohen-Macaulay rings. Stud. Appl. Math. 54, 135-142 (1975)
10. Stanley, R.P.: Cohen-Macaulay complexes. In: Aigner, M. (ed.) Higher Combinatorics, pp. 51-62. Reidel, Dordrecht/Boston (1977)
11. Stanley, R.P.: Balanced Cohen-Macaulay complexes. Trans. Am. Math. Soc. 249, 139-157 (1979)
12. Stanley, R.P.: Enumerative Combinatorics, vol.
1. Wadsworth & Brooks/Cole, Pacific Grove (1986). Second printing, Cambridge Studies in Advanced Mathematics, vol.
49. Cambridge University Press, Cambridge (1997)
13. Stanley, R.P.: Generalized h-vectors, intersection cohomology of toric varieties and related results. In: Nagata, N., Matsumura, H. (eds.) Commutative Algebra and Combinatorics. Adv. Stud. Pure Math., vol. 11, pp. 187-213. Kinokuniya/North-Holland, Tokyo/Amsterdam/New York (1987)
14. Stanley, R.P.: f -vectors and h-vectors of simplicial posets. J. Pure Appl. Algebra 71, 319-331 (1991)
15. Stanley, R.P.: Combinatorics and Commutative Algebra, 2nd edn. Birkhäuser, Basel (1996) J Algebr Comb (2012) 35:373-388
16. Stanley, R.P.: Positivity problems and conjectures in algebraic combinatorics. In: Arnold, V., Atiyah, M., Lax, P., Mazur, B. (eds.) Mathematics: Frontiers and Perspectives, pp. 295-319. Amer. Math. Society, Providence (2000)
17. Walker, J.: Canonical homeomorphisms of posets. Eur. J. Comb. 9, 97-107 (1988)
18. Yanagawa, K.: Higher Cohen-Macaulay property of squarefree modules and simplicial posets. Proc. Am. Math. Soc. 139, 3057-3066 (2011).
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition