The f- and h-vectors of interval subdivisions

Abstract

We show that the γ-vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In particular, we prove that such γ-vector is the f -vector of some balanced simplicial complex. Moreover, we show that the local γ-vector of the interval subdivision of a simplex is nonnegative; answering a question by Juhnke-Kubitzke et al.Conjecture 1.1 is a strengthening of the well known Charney-Devis conjecture. The Gal conjecture holds for all Coxeter complexes (see [Ste08]), for the dual simplicial complexes of associahedron and cyclohedron (see [NP11]), and for barycentric subdivision of homology sphere (see[NPT11]). The authors in [NP11] conjectured further strengthening of Gal conjecture. Conjecture 1.2. [NP11, Problem 6.4] If ∆ is a flag homology sphere then γ(∆) is the f -vector of some balanced simplicial complex.This conjecture holds for the dual simplicial complex of all flag nestohedera, see in [Ais14]. Frohmader [Fro08, Theorem 1.1] showed that the f -vector of any flag simplicial complex satisfies the Frankl-Füredi-Kalai (FFK) inequalities (see [FFK88]). In [NPT11], authors showed that the γ-vector of the barycentric subdivision of a homology sphere satisfies the FFK inequalities, i.e., the f -vector of a balanced simplicial complex. The first aim of this paper is the confirmation of Conjecture 1.2 in the case of the interval subdivision of a homology sphere. The main theorem is stated as: Theorem 1.3. If ∆ is a simplicial complex with a nonnegative and symmetric h-vector, then the γ-vector of the interval subdivision of ∆ is the f -vector of a balanced simplicial complex.

Symmetric Decompositions, Triangulations and Real‐rootedness

“…, r − 1}. For r = 2 they have been considered before in [4] (see, for instance, Corollary 4.7 there). PROPOSITION 5.5.…”

On $$\gamma $$- and local $$\gamma $$-vectors of the interval subdivision

“…The interval subdivision Int( ) of a simplicial complex is the simplicial complex on the vertex set I ( \∅), where I ( \∅) := {[A, B] | ∅ = A ⊆ B ∈ } as a partially ordered set ordered by inclusion defined as Walker [16], the simplicial complex of all chains in the partially ordered set I ( \ ∅) is a subdivision of . In [2], the authors have given a formula for the f -vector of a simplicial complex under the interval subdivision in the following theorem. Theorem 2.1 [2, Theorem 2.2] Let be a (d − 1)-dimensional simplicial complex.…”

“…In [2], the authors have given a combinatorial description of the h-vector of a simplicial complex under the interval subdivision in terms of these Eulerian numbers.…”

“…This work is based on the study of certain refinement of Eulerian numbers of type B used by the present authors in [2] to describe the h-vector of the interval subdivision Int( ) of a simplicial complex . The interval subdivision of a simplicial complex was introduced by Walker in [16].…”

“…By Walker [16,Theorem 6.1(a)], the simplicial complex of all chains in the partially ordered set I ( \ ∅) := {[A, B] | ∅ = A ⊆ B ∈ } is a subdivision of . In [2], the authors have given the combinatorial description of the f -and h-vectors of a simplicial complex under the interval subdivision. The second aim of this paper is to study the local γ -vector of the interval subdivision of a simplex.…”