# ALGEBRAIC COMBINATORICS

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 $\Delta$ of dimension $d-1$ satisfies ${h}_{i}\left(\Delta \right)=\left(\genfrac{}{}{0pt}{}{d}{i}\right)$ for some $i\ge 1$, then ${h}_{j}\left(\Delta \right)=\left(\genfrac{}{}{0pt}{}{d}{j}\right)$ for all $j\ge i$. Much more recently, Klee, Nevo, Novik, and Zheng conjectured that if a centrally symmetric simplicial polytope $P$ of dimension $d$ satisfies ${g}_{i}\left(\partial P\right)=\left(\genfrac{}{}{0pt}{}{d}{i}\right)-\left(\genfrac{}{}{0pt}{}{d}{i-1}\right)$ for some $d/2\ge i\ge 1$, then ${g}_{j}\left(\partial P\right)=\left(\genfrac{}{}{0pt}{}{d}{j}\right)-\left(\genfrac{}{}{0pt}{}{d}{j-1}\right)$ for all $d/2\ge j\ge i$. This note uses stress spaces to prove both of these conjectures.

Revised:
Accepted:
Published online:
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/

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