Combinatorics Seminar: "Exponential generating functions and geometry"
Abstract: This talk is an exploration of recent results on exponential Hilbert series in two different areas: the geometry of complex flag varieties and the Stanley-Reisner ring of a simplicial complex. We develop closed formulas for both situations and explore the geometric and algebraic data encoded in each formula. For a partial flag variety, we find that the exponential Hilbert series encodes similar data to the classical Hilbert series along with some additional algebraic data. For a simplicial complex, the coarsely-graded exponential series determines both the face vector, and (through the face vector) the h-vector. We present a simple condition on the exponential Hilbert series equivalent to the Dehn-Sommerville equations.