Flag complex face structures and decompositions

Abstract

One of the most common and effective methods of obtaining structural information on simplicial complexes is to use tools from algebraic geometry/commutative algebra (often motivated by properties of toric varieties). However, there is no general algebro-geometric description of components of the gamma vector holding for arbitary flag simplicial spheres. This invariant occurs in many different contexts including permutation statistics, signatures of toric varieties, and Euler characteristics of nonpositively curved piecewise Euclidean manifolds. Combinatorial methods resulting from an explicit inverted Chebyshev expansion give rise to new positivity properties and cell complex structures that are of interest in their own right. Note that the focus is on the $f$-vector rather than the $h$-vector in ``algebraic'' settings. For flag simplicial spheres $Δ$, the fact that $h(Δ) = f(Γ)$ and compatibility between Chebyshev expansions and a modification of the $f$-polynomial by work of Hetyei are the key inputs. In the main formula implying new positivity results, local structures of $CAT(0)$ complexes and cubical analogues of barycentric subdivisions give deeper connections with cubical complex structures complementing earlier work related to the top gamma vector component. Afterwards, we return to the motivating example of barycentric subdivisions and consider how $f$-vectors of Cohen--Macaulay and vertex decomposable flag complexes in geometric settings decompose and interact with geometric transformations. This includes subdivisions of simplicial complexes and recursive properties they share with vertex decomposable flag complexes.