报告人：杨立波 教授 （南开大学）
报告摘要：Brenti conjectured that, for any finite Coxeter group, the descent generating polynomial has only real zeros, and he left the type D case open. Dilks, Petersen, and Stembridge proposed a companion conjecture, which states that, for any irreducible finite Weyl group, the affine descent generating polynomial has only real zeros, and they left the type B and type D cases open. By developing the theory of s-Eulerian polynomials, Savage and Visontai confirmed the type D case of the former conjecture and the type B case of the latter conjecture. In this paper we give an analytic approach to these two combinatorial conjectures. In particular, based on the Hermite--Biehler theorem and the theory of linear transformations preserving Hurwitz stability, we obtain the Hurwitz stability of certain polynomials related to the descent generating polynomials of type D, and thus give an alternative proof of Savage and Visontai's results. This new approach also enables us to prove Hyatt's conjectures on the interlacing property of half Eulerian polynomials of type B and type D, and to prove that the $h$-polynomial of certain subcomplexes of Coxeter complexes of type D has only real zeros. We further study the Hurwitz stability of certain polynomials related to the affine descent generating polynomials of type D, and completely confirm Dilks, Petersen, and Stembridge's conjecture. This is a joint work with Philip Zhang.