On the Charney-Davis and Neggers-Stanley Conjectures

The title refers to two seemingly unrelated open combinatorial conjectures from the late 1980's/early 1990's. The Charney-Davis Conjecture asserts an inequality on face numbers of certain simplicial spheres. The Neggers-Stanley Conjecture asserts that certain generating functions (counting linear extensions of a poset by number of descents) have only real zeroes.

After reviewing both conjectures, we will give results showing that these two conjectures are closely related. We then speculate that the right context in which to think about both may be the interaction of Koszul rings with the theory of total positivity and Polya frequency sequences.

This is joint work with Volkmar Welker.