Subdivisions of lower Eulerian posets and KLS theory
Alan Stapledon
TL;DR
The work develops a unified, incidence-algebra framework linking local $h$-polynomials of strong formal subdivisions of lower Eulerian posets to Kazhdan-Lusztig-Stanley (KLS) invariants under a canonical bijection with triples $(\Gamma,\rho_\Gamma,q)$. It proves explicit formulas connecting the global KLS data of the Cyl$(\sigma)$ construction to the subdivision data through the operator $\Delta$ applied to $\ell_\sigma$, and shows how Braden–MacPherson’s relative $g$-polynomials encode these local polynomials, with natural equivariant and Ehrhart-theoretic extensions. The paper further extends the theory to equivariant KLS, providing fixed-point formulas and a robust framework for equivariant Ehrhart theory, including polynomiality criteria and practical decompositions in terms of faces. Overall, it deepens the interplay between subdivision combinatorics, poset KLS invariants, and geometric-context invariants, enabling new tools for studying subdivisions, equivariant phenomena, and lattice-point enumeration in a unified setting.
Abstract
In a companion paper, a canonical bijection was established between strong formal subdivisions of lower Eulerian posets and triples consisting of a lower Eulerian poset, a corresponding rank function, and a non-minimal element such that the join with any other element exists. The main goal of this paper is to relate the local $h$-polynomials of a strong formal subdivision to the Kazhdan-Lusztig-Stanley (KLS) invariants associated to its corresponding lower Eulerian poset under this bijection. As an application, we show that Braden and MacPherson's relative $g$-polynomials are alternative encodings of corresponding local $h$-polynomials. We also further develop equivariant KLS theory and give equivariant generalizations of our main results, as well as an application to equivariant Ehrhart theory.