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Flag complex face structures and decompositions

Soohyun Park

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.

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 -vector rather than the -vector in ``algebraic'' settings. For flag simplicial spheres , the fact that and compatibility between Chebyshev expansions and a modification of the -polynomial by work of Hetyei are the key inputs. In the main formula implying new positivity results, local structures of 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 -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.
Paper Structure (6 sections, 27 theorems, 58 equations)

This paper contains 6 sections, 27 theorems, 58 equations.

Table of Contents

  1. Geometry of the gamma vector as a whole via Tchebyshev transformations and subdivisions
  2. Background on Tchebyshev transforms/subdivisions and compatibility
  3. Cell complex structures induced by the $f$-vector perspective
  4. Background related to cell and cubical complexes
  5. Colorings, Tchebyshev subdivisions, and intervals
  6. Recursive properties involving the $f$-vector perspective based on older decompositions

Key Result

Theorem 1

(Gamma vectors and inverted Chebyshev expansions, Theorem 1.7 on p. 7 of Pgamcheb) Let $h(t) = h_0 + h_1 t + \ldots + h_{d - 1} t^{d - 1} + h_d t^d$ be a reciprocal polynomial satisfying the relations $h_k = h_{d - k}$ for $0 \le k \le \frac{d}{2}$. Then, the gamma-polynomial associated to $h(t)$ is where This is a sort of "inverted Chebyshev basis expansion". Alternatively, we can rewrite this a

Theorems & Definitions (58)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Proposition 5
  • Theorem 6
  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Remark 1.4
  • ...and 48 more