On the Charney-Davis and Neggers-Stanley conjectures

For a graded naturally labelled poset P, it is shown that the P-Eulerian polynomial A formula is presented counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the Neggers-Stanley conjecture on real zeroes for W (P, t) in these cases. The result is deduced from McMullen's g-Theorem, by exhibiting a simplicial polytopal sphere whose h-polynomial is W (P, t). Whenever this simplicial sphere turns out to be flag, that is, its minimal non-faces all have cardinality two, it is shown that the Neggers-Stanley Conjecture would imply the Charney-Davis Conjecture for this sphere. In particular, it is shown that the sphere is flag whenever the poset P has width at most 2. In this case, the sphere is shown to have a stronger geometric property (local convexity), which then implies the Charney-Davis Conjecture in this case via a result from Leung and Reiner (Duke Math. J. 111 (2002) 253). It is speculated that the proper context in which to view both of these conjectures may be the theory of Koszul algebras, and some evidence is presented.