The stresses on centrally symmetric complexes and the lower bound theorems

Novik, Isabella; Zheng, Hailun

doi:10.5802/alco.168

Novik, Isabella ¹; Zheng, Hailun ²

¹ Department of Mathematics University of Washington Seattle, WA 98195-4350, USA
² Department of Mathematical Sciences University of Copenhagen Universitesparken 5, 2100 Copenhagen, Denmark

Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 541-549.

Abstract

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

Received: 2020-08-28
Revised: 2021-01-10
Accepted: 2021-01-11
Published online: 2021-06-22

DOI: 10.5802/alco.168

Classification: 05E45, 05E40, 13F55, 52B05, 52B15
Keywords: Cohen–Macaulay complexes, polytopes, centrally symmetric, face numbers, stress spaces.

Author's affiliations:

Novik, Isabella ¹; Zheng, Hailun ²

¹ Department of Mathematics University of Washington Seattle, WA 98195-4350, USA
² Department of Mathematical Sciences University of Copenhagen Universitesparken 5, 2100 Copenhagen, Denmark

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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/

References
Cited by

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

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