Lorenzo Vecchi: Chow polynomials of totally nonnegative matrices and posets

Abstract: We study Chow polynomials of posets, a combinatorial generalization of the Hilbert series of the Chow ring of a matroid. We introduce Chow polynomials of lower-triangular matrices (with ones on the diagonal), via a generalization of the well-known relation between derangement and Eulerian polynomials. Our main result shows that if such a matrix is totally nonnegative, then its Chow polynomial is real-rooted. This establishes new real-rootedness results for a broad class of posets we call totally nonnegative posets, which contains projective and affine geometries, face lattices of cubical complexes, dual partition lattices, and confirms the real-rootedness conjecture for paving matroids and their lattices of flats. Our methods employ new interlacing techniques and a generalization of the deranged map by Brändén and Solus.

This is based on joint work with Petter Brändén.