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$f$-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions

Soohyun Park

Abstract

We show that there are $f$-vectors of balanced simplicial complexes giving a source of simplicial complexes exhibiting a Boolean decomposition similar to a geometric Lefschetz decomposition. The objects we are working with are $h$-vectors of flag spheres and balanced simplicial complexes whose $f$-vectors are equal to them. This builds on work of Nevo--Petersen--Tenner on a conjecture of Nevo--Petersen that the gamma vector of an odd-dimensional flag sphere is the $f$-vector of a balanced simplicial complex (which was shown for barycentric subdivisions by Nevo--Petersen--Tenner). We can connect our decomposition to positivity questions on reciprocal/palindromic polynomials associated to flag spheres and geometric questions motivating them. In addition, we note that the degrees in the Lefschetz-like decomposition are not halved unlike the usual $h$-vector setting.

$f$-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions

Abstract

We show that there are -vectors of balanced simplicial complexes giving a source of simplicial complexes exhibiting a Boolean decomposition similar to a geometric Lefschetz decomposition. The objects we are working with are -vectors of flag spheres and balanced simplicial complexes whose -vectors are equal to them. This builds on work of Nevo--Petersen--Tenner on a conjecture of Nevo--Petersen that the gamma vector of an odd-dimensional flag sphere is the -vector of a balanced simplicial complex (which was shown for barycentric subdivisions by Nevo--Petersen--Tenner). We can connect our decomposition to positivity questions on reciprocal/palindromic polynomials associated to flag spheres and geometric questions motivating them. In addition, we note that the degrees in the Lefschetz-like decomposition are not halved unlike the usual -vector setting.

Paper Structure

This paper contains 5 sections, 11 theorems, 20 equations.

Table of Contents

  1. Background
  2. $f$-vectors of balanced simplicial complexes and Boolean decompositions
  3. Boolean decompositions and $f$-vectors of balanced simplicial complexes
  4. 1-dimensional case and local to global constructions
  5. Connections to geometric Lefschetz decompositions

Key Result

Theorem 7

(Nevo--Petersen--Tenner, Theorem 1.1 on p. 1365 of NPT) If $\Delta$ is the barycentric subdivision of a simplicial complex, then $\gamma(\Delta)$ is the $f$-vector of a balanced simplicial complex.

Theorems & Definitions (24)

  • Definition 1
  • Conjecture 2
  • Definition 3
  • Conjecture 4
  • Definition 5
  • Conjecture 6
  • Theorem 7
  • Proposition 8
  • Theorem 9
  • Definition 1.1
  • ...and 14 more