On the Charney-Davis and Neggers-Stanley conjectures
Besides addressing the issues noted in the title of this wide-ranging paper, it also establishes an overall context in which a number of results of interest may find themselves proven or strengthened as conjectures. For the Neggers-Stanley conjecture this is particularly interesting since its ``overthrow'' in particular forms naturally has produced a fall-out of other questions and conjectures which add to the overall activity which may be considered relevant to the classes of problems they seek to deal with. Thus, the authors speculate impressively that a number of problems may be addressed in the context of Koszul Algebras and Pólya Frequency (PF) Sequences, where the interactive medium may be that of the Stanley-Reisner ring (algebra) of the order complex associated with some particular type of poset for which the Neggers-Stanley (or some other, e.g., the Charney-Davis) conjecture is to be verified. The machinery though ``usual'' is highly complex and for any newcomer a warning might be useful that for an insight into the technical issues at stake, proper preparation is a sine-qua-non even if the fundamental statements upon which this subject focuses, whether in the theory of posets, digraphs, or elsewhere, are rather much easier to understand. What the authors indicate is the existence of a larger context into which these problems fit, not forgetting that even among posets there are those which are representatives of another domain not includable in this nevertheless very substantial universe delineated so well by the authors of this very interesting paper.
scientific article; zbMATH DE number 2153226
scientific article