Balanced Cohen-Macaulay complexes

Stanley, Richard P.

doi:10.1090/S0002-9947-1979-0526314-6

Balanced Cohen-Macaulay complexes
HTML articles powered by AMS MathViewer

by Richard P. Stanley

Trans. Amer. Math. Soc. 249 (1979), 139-157

DOI: https://doi.org/10.1090/S0002-9947-1979-0526314-6

PDF | Request permission

Abstract:

A balanced complex of type $({a_1},\ldots ,{a_m})$ is a finite pure simplicial complex $\Delta$ together with an ordered partition $({V_1},\ldots ,{V_m})$ of the vertices of $\Delta$ such that card$({V_i} \cap F) = {a_i}$, for every maximal face F of $\Delta$. If ${\mathbf {b}} = ({b_1},\ldots ,{b_m})$, then define ${f_\textbf {b}}(\Delta )$ to be the number of $F \in \Delta$ satisfying card$({V_i} \cap F) = {b_i}$. The formal properties of the numbers ${f_\textbf {b}}(\Delta )$ are investigated in analogy to the f-vector of an arbitrary simplicial complex. For a special class of balanced complexes known as balanced Cohen-Macaulay complexes, simple techniques from commutative algebra lead to very strong conditions on the numbers${f_\textbf {b}}(\Delta )$. For a certain complex $\Delta (P)$ coming from a poset P, our results are intimately related to properties of the Möbius function of P.

References

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
G. F. Clements, More on the generalized Macaulay theorem, Discrete Math. 1 (1971/72), no. 3, 247–255. MR 285406, DOI 10.1016/0012-365X(71)90013-6
G. F. Clements and B. Lindström, A generalization of a combinatorial theorem of Macaulay, J. Combinatorial Theory 7 (1969), 230–238. MR 246781
Gopal Danaraj and Victor Klee, Which spheres are shellable?, Ann. Discrete Math. 2 (1978), 33–52. MR 500687

An amusing reformulation of the Four Color Problem

24

T. Gallai, Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar. 18 (1967), 25–66 (German). MR 221974, DOI 10.1007/BF02020961
Alain Ghouila-Houri, Caractérisation des graphes non orientés dont on peut orienter les arětes de manière à obtenir le graphe d’une relation d’ordre, C. R. Acad. Sci. Paris 254 (1962), 1370–1371 (French). MR 172275
P. C. Gilmore and A. J. Hoffman, A characterization of comparability graphs and of interval graphs, Canadian J. Math. 16 (1964), 539–548. MR 175811, DOI 10.4153/CJM-1964-055-5
Curtis Greene and Daniel J. Kleitman, Proof techniques in the theory of finite sets, Studies in combinatorics, MAA Stud. Math., vol. 17, Math. Assoc. America, Washington, D.C., 1978, pp. 22–79. MR 513002
Melvin Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975) Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977, pp. 171–223. MR 0441987
G. Katona, A theorem of finite sets, Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic Press, New York, 1968, pp. 187–207. MR 0290982
Joseph B. Kruskal, The number of simplices in a complex, Mathematical optimization techniques, Univ. California Press, Berkeley, Calif., 1963, pp. 251–278. MR 0154827

Some properties of enumeration in the theory of modular systems

26

P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika 17 (1970), 179–184. MR 283691, DOI 10.1112/S0025579300002850
James R. Munkres, Topological results in combinatorics, Michigan Math. J. 31 (1984), no. 1, 113–128. MR 736476, DOI 10.1307/mmj/1029002969
Gerald Allen Reisner, Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), no. 1, 30–49. MR 407036, DOI 10.1016/0001-8708(76)90114-6
Gian-Carlo Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368 (1964). MR 174487, DOI 10.1007/BF00531932
Richard P. Stanley, Ordered structures and partitions, Memoirs of the American Mathematical Society, No. 119, American Mathematical Society, Providence, R.I., 1972. MR 0332509
R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197–217. MR 309815, DOI 10.1007/BF02945028
Richard P. Stanley, Combinatorial reciprocity theorems, Advances in Math. 14 (1974), 194–253. MR 411982, DOI 10.1016/0001-8708(74)90030-9
Richard P. Stanley, Finite lattices and Jordan-Hölder sets, Algebra Universalis 4 (1974), 361–371. MR 354473, DOI 10.1007/BF02485748
Richard P. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Studies in Appl. Math. 54 (1975), no. 2, 135–142. MR 458437, DOI 10.1002/sapm1975542135
Richard P. Stanley, Cohen-Macaulay complexes, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 31, Reidel, Dordrecht-Boston, Mass., 1977, pp. 51–62. MR 0572989
Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57–83. MR 485835, DOI 10.1016/0001-8708(78)90045-2

Decompositions, shellings, and diameters of simplicial complexes and convex polyhedra