Recursive properties of Cohen--Macaulay flag simplicial complexes and Lefschetz decompositions from $f$-vectors

TL;DR

This paper investigates the recursive and Lefschetz properties of flag simplicial complexes, revealing explicit Boolean decompositions of their $f$-vectors through specific transformations, and connecting these to algebraic maps satisfying Lefschetz-like conditions.

Contribution

It introduces explicit transformations and decompositions for flag simplicial complexes' $f$-vectors, linking combinatorial and algebraic Lefschetz properties.

Findings

Explicit Boolean decompositions of $f$-vectors via transformations.

Connections between local decompositions and Lefschetz-type maps.

Balanced complexes suggest algebraic Lefschetz maps.

Abstract

Most applications of the hard Lefschetz theorem related to combinatorial properties of simplicial complexes involve their $h$-vectors. In the context of positivity properties involving $h$-vectors of flag spheres, $f$-vectors with a Lefschetz-type ``Boolean'' decomposition have been studied. In this note, we explore families of flag simplicial complexes where we can see this Boolean decomposition explicitly in terms of transformations connecting different simplicial complexes in this family. Note that we will take complexes in a given dimension to be PL homeomorphic to each other. In particular, the existence of a Boolean decomposition patched from local parts can be phrased in terms of a certain map formally satisfying an analogue of the hard Lefschetz theorem. The map is given by the composition of a double suspension with a ``net single edge subdivision''. Here, the former contributes to the Boolean part and the latter contributes to the disjoint non-Boolean part. The fact that the simplicial complex with the given $f$-vector can be taken to be balanced suggests algebraic versions of maps connected to these decompositions.