The Lefschetz property for barycentric subdivisions of shellable complexes

TL;DR

This paper proves an 'almost strong Lefschetz' property for barycentric subdivisions of shellable complexes, leading to new results on the unimodality and $g$-vector properties, and verifying the $g$-conjecture for barycentric subdivisions of homology spheres.

Contribution

It establishes an algebraic Lefschetz property for barycentric subdivisions of shellable complexes, confirming conjectures and deriving new combinatorial inequalities.

Findings

Unimodality of the $h$-vector for barycentric subdivisions of Cohen-Macaulay complexes.

Verification of the $g$-conjecture for barycentric subdivisions of homology spheres.

New inequalities on Eulerian statistics related to permutations.

Abstract

We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its $g$-vector is an $M$-sequence. In particular, the (combinatorial) $g$-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of 1.