Face vectors of subdivided simplicial complexes

Abstract

Brenti and Welker have shown that, for any ( d − 1 ) -dimensional simplicial complex X , the f -vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X . We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number −2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric motivation to generalize the combinatorial statements to arbitrary subdivision methods: any subdivision method will exhibit similar limit behavior and symmetry. Our techniques allow us to compute explicit formulas for the values of the limit roots in the case of barycentric subdivision.