Shellable complexes from multicomplexes
Jonathan Browder
DOI: 10.1007/s10801-009-0206-z
Abstract
Suppose a group G acts properly on a simplicial complex Γ . Let l be the number of G-invariant vertices, and p 1, p 2,\cdots , p m be the sizes of the G-orbits having size greater than 1. Then Γ must be a subcomplex of \varLambda = \varDelta l -1* {\P}\varDelta p 1 -1 * \frac{1}{4} * {\P}\varDelta p m -1 \varLambda=\varDelta ^{l-1}*
tial \varDelta ^{p_{1}-1} *\cdots*
tial \varDelta ^{p_{m}-1}. A result of Novik gives necessary conditions on the face numbers of Cohen-Macaulay subcomplexes of Λ . We show that these conditions are also sufficient, and thus provide a complete characterization of the face numbers of these complexes.
tial \varDelta ^{p_{1}-1} *\cdots*
tial \varDelta ^{p_{m}-1}. A result of Novik gives necessary conditions on the face numbers of Cohen-Macaulay subcomplexes of Λ . We show that these conditions are also sufficient, and thus provide a complete characterization of the face numbers of these complexes.
Pages: 99–112
Keywords: keywords simplicial complex; $f$-vector; multicomplex
Full Text: PDF
References
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2. Björner, A., Frankl, P., Stanley, R.: The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem. Combinatorica 7(1), 23-34 (1987)
3. Browder, J., Novik, I.: Face numbers of generalized balanced Cohen-Macaulay complexes.
4. Clements, G.F., Lindström, B.: A generalization of a combinatorial theorem of Macaulay. J. Comb. Theory 7, 230-238 (1969) J Algebr Comb (2010) 32: 99-112
5. Katona, G.: A theorem of finite sets. In: Theory of Graphs, Proc. Colloq., Tihany, 1966, pp. 187-207. Academic Press, New York (1968)
6. Kruskal, J.B.: The number of simplices in a complex. In: Mathematical Optimization Techniques, pp. 251-278. Univ. of California Press, Berkeley (1963)
7. McMullen, P., Shephard, G.C.: Convex Polytopes and the Upper Bound Conjecture. Cambridge University Press, London (1971). Prepared in collaboration with J.E. Reeve and A.A. Ball, London Mathematical Society Lecture Note Series, 3
8. Novik, I.: On face numbers of manifolds with symmetry. Adv. Math. 192(1), 183-208 (2005)
9. Stanley, R.P.: Cohen-Macaulay complexes. In: Higher Combinatorics, Proc. NATO Advanced Study Inst., Berlin,
1976. NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci., vol. 31, pp. 51-62. Reidel, Dordrecht (1977)
10. Stanley, R.P.: On the number of faces of centrally-symmetric simplicial polytopes. Graphs Comb.
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