On the Charney-Davis and Neggers-Stanley conjectures

Abstract. For a graded naturally labelled poset P, it is shown that the P-Eulerian polynomial W(P, t): = � t des(w) w∈L(P) 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 hpolynomial 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 (locally convexity), which then implies the Charney-Davis Conjecture in this case via a result from [27]. 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. 1