Balanced generalized lower bound inequality for simplicial polytopes
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A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the h-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial d-polytopes, that is simplicial d-polytopes whose underlying graphs are d-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their h-numbers.
Mathematics Subject Classification52B05 13F55 05C15
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The first author was partially supported by DFG GK-1916. The second author was partially supported by JSPS KAKENHI 25400043. We would like to thank Steven Klee and Isabella Novik for their helpful comments on the paper.
- 1.Adiprasito, K.: Toric chordality. J. Math. Pures Appl. (to appear)Google Scholar
- 2.Bayer, M., Billera, L.: Counting faces and chains in polytopes and posets. Combinatorics and Algebra (Boulder, Colo., 1983), Contemp. Math., vol. 34, Amer. Math. Soc. (1984)Google Scholar
- 5.Harima, T., Maeno, T., Morita, H., Numata, Y., Wachi, A., Watanabe, J.: The Lefschetz Properties. Lecture Notes in Mathematics, vol. 2080. Springer (2013)Google Scholar
- 13.Swartz, E.: Lower bounds for \(h\)-vectors of \(k\)-CM, independence and broken circuit complexes. SIAM J. Discrete Math. 18, 647–661 (2004/2005)Google Scholar