Flag subdivisions and γ-vectors

Christos A. Athanasiadis

Vol. 259 (2012), No. 2, 257–278

DOI: 10.2140/pjm.2012.259.257

Abstract

The γ-vector is an important enumerative invariant of a flag simplicial homology sphere. It has been conjectured by Gal that this vector is nonnegative for every such sphere Δ and by Reiner, Postnikov and Williams that it increases when Δ is replaced by any flag simplicial homology sphere that geometrically subdivides Δ. Using the nonnegativity of the γ-vector in dimension 3, proved by Davis and Okun, as well as Stanley’s theory of simplicial subdivisions and local h-vectors, the latter conjecture is confirmed in this paper in dimensions 3 and 4.

Keywords

flag complex, homology sphere, simplicial subdivision, flag subdivision, face enumeration, γ-vector

Mathematical Subject Classification 2010

Primary: 05E45

Secondary: 05E99

Milestones

Received: 11 January 2012

Revised: 28 May 2012

Accepted: 4 June 2012

Published: 3 October 2012