The stresses on centrally symmetric complexes and the lower bound theorems
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 541-549.

In 1987, Stanley conjectured that if a centrally symmetric Cohen–Macaulay simplicial complex Δ of dimension d-1 satisfies h i (Δ)=d i for some i1, then h j (Δ)=d j for all ji. Much more recently, Klee, Nevo, Novik, and Zheng conjectured that if a centrally symmetric simplicial polytope P of dimension d satisfies g i (P)=d i-d i-1 for some d/2i1, then g j (P)=d j-d j-1 for all d/2ji. This note uses stress spaces to prove both of these conjectures.

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DOI: https://doi.org/10.5802/alco.168
Classification: 05E45,  05E40,  13F55,  52B05,  52B15
Keywords: Cohen–Macaulay complexes, polytopes, centrally symmetric, face numbers, stress spaces.
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Novik, Isabella; Zheng, Hailun. The stresses on centrally symmetric complexes and the lower bound theorems. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 541-549. doi : 10.5802/alco.168. https://alco.centre-mersenne.org/articles/10.5802/alco.168/

[1] Adiprasito, Karim Combinatorial Lefschetz theorems beyond positivity (https://arxiv.org/abs/1812.10454)

[2] Billera, Louis J.; Lee, Carl W. A proof of the sufficiency of McMullen’s conditions for f-vectors of simplicial convex polytopes, J. Combin. Theory Ser. A, Volume 31 (1981) no. 3, pp. 237-255 | Article | MR 635368 | Zbl 0479.52006

[3] Björner, Anders; Frankl, Peter; Stanley, Richard P. The number of faces of balanced Cohen–Macaulay complexes and a generalized Macaulay theorem, Combinatorica, Volume 7 (1987) no. 1, pp. 23-34 | Article | MR 905148 | Zbl 0651.05010

[4] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995, xvi+785 pages | Article | MR 1322960 | Zbl 0819.13001

[5] Hochster, Melvin 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.), Volume 26, Dekker, 1977, pp. 171-223 | MR 0441987 | Zbl 0351.13009

[6] Klee, Steven; Nevo, Eran; Novik, Isabella; Zheng, Hailun A lower bound theorem for centrally symmetric simplicial polytopes, Discrete Comput. Geom., Volume 61 (2019) no. 3, pp. 541-561 | Article | MR 3918547 | Zbl 1431.52017

[7] Lee, Carl W. P.L.-spheres, convex polytopes, and stress, Discrete Comput. Geom., Volume 15 (1996) no. 4, pp. 389-421 | Article | MR 1384883 | Zbl 0856.52009

[8] Reisner, Gerald A. Cohen–Macaulay quotients of polynomial rings, Adv. Math., Volume 21 (1976) no. 1, pp. 30-49 | Article | MR 0407036 | Zbl 0345.13017

[9] Stanley, Richard P. The upper bound conjecture and Cohen–Macaulay rings, Studies in Appl. Math., Volume 54 (1975) no. 2, pp. 135-142 | Article | MR 458437 | Zbl 0308.52009

[10] Stanley, Richard P. Cohen–Macaulay complexes, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) (NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci.), Volume 31 (1977), pp. 51-62 | MR 0572989 | Zbl 0376.55007

[11] Stanley, Richard P. Balanced Cohen–Macaulay complexes, Trans. Amer. Math. Soc., Volume 249 (1979) no. 1, pp. 139-157 | Article | MR 526314 | Zbl 0411.05012

[12] Stanley, Richard P. The number of faces of a simplicial convex polytope, Adv. in Math., Volume 35 (1980) no. 3, pp. 236-238 | Article | MR 563925 | Zbl 0427.52006

[13] Stanley, Richard P. On the number of faces of centrally-symmetric simplicial polytopes, Graphs Combin., Volume 3 (1987) no. 1, pp. 55-66 | Article | MR 932113 | Zbl 0611.52002

[14] Stanley, Richard P. A monotonicity property of h-vectors and h * -vectors, European J. Combin., Volume 14 (1993) no. 3, pp. 251-258 | Article | MR 1215335 | Zbl 0799.52008

[15] Stanley, Richard P. Combinatorics and commutative algebra, Progress in Mathematics, 41, Birkhäuser Boston, Inc., Boston, MA, 1996, x+164 pages | MR 1453579 | Zbl 0838.13008

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