Combinatorial g-conjecture for interval subdivisions

Abstract

We verify the g-conjecture for interval subdivisions of Cohen–Macaulay simplicial complexes, using purely combinatorial methods. More precisely, we show that the g-vector of the interval subdivision of a Cohen–Macaulay simplicial complex is an f-vector. Murai defined the $g″$-vector, starting from the $h″$-vector introduced by Novik for Buchsbaum simplicial complexes. We prove that the $g″$-vector of the interval subdivision of a Buchsbaum simplicial complex is an f-vector of some simplicial complex.

Keywords:

• g-Vector
• h-vector
• simplicial complex
• subdivision of a simplicial complex

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 05E45
• 52B05
• 05A05

Acknowledgments

We are deeply indebted to anonymous referee for his/her comments and suggestions based on careful observations on the earlier version, which improved this paper in a great deal.