Weighted Ehrhart theory via equivariant toric geometry
Laurenţiu Maxim, Jörg Schürmann
TL;DR
This work builds a geometric, toric-geometry framework for generalized weighted Ehrhart theory by interpreting face-weighted lattice point sums as equivariant Hodge-Chern data of mixed Hodge modules on the toric variety $X_P$. It introduces twisted equivariant Hodge-Chern classes $\mathrm{DR}^{\mathbb T}_y$ and shows how weighted Ehrhart polynomials $E^\varphi_{P,f}(\ell,y)$ arise from equivariant Euler characteristics, establishing reciprocity and duality across faces and connecting to Stanley's $g$-polynomials via polar duality. The authors unify and extend Brion–Vergne reciprocity with purity/dulity phenomena, provide explicit toric formulae for orbit-closures $V_\sigma$, and demonstrate how these geometric invariants encode generalized lattice-point sums, including special cases that recapture known combinatorial identities. The framework has potential to deepen the interplay between topology of toric varieties, equivariant Hodge theory, and lattice-point enumeration, with concrete computational expressions for generalized weighted Ehrhart polynomials.
Abstract
We give a $K$-theoretic and geometric interpretation for a generalized weighted Ehrhart theory of a full-dimensional lattice polytope $P$, depending on a given homogeneous polynomial function $\varphi$ on $P$, and with Laurent polynomial weights $f_Q(y)\in \mathbb{Z}[y^{\pm 1}]$ associated to the faces $Q \preceq P$ of the polytope. For this purpose, we calculate equivariant $K$-theoretic Hodge-Chern classes of a torus-equivariant mixed Hodge module $\mathcal{M}$ on the toric variety $X_P$ associated to $P$. For any integer $\ell$, we introduce an equivariant Hodge $χ_y$-polynomial $χ_y(X_P, \ell D_P; [\mathcal{M}])$, with $D_P$ the corresponding ample Cartier divisor on $X_P$ (defined by the facet presentation of $P$). Motivic properties of the Hodge-Chern classes are used to express this equivariant Hodge polynomial in terms of weighted character sums fitting with a generalized weighted Ehrhart theory. The equivariant Hodge polynomials are shown to satisfy a reciprocity and purity formula fitting with the duality for equivariant mixed Hodge modules, and implying similar properties for the generalized weighted Ehrhart polynomials. In the special case of the equivariant intersection cohomology mixed Hodge module, with the weight function given by Stanley's $g$-function of the polar polytope of $P$, we recover in geometric terms a recent combinatorial formula of Beck-Gunnells-Materov. More generally, motivated by the analogy to the Kazhdan-Lusztig theory, we introduce a duality involution on the free $ \mathbb{Z}[y^{\pm 1}]$-module of weight functions corresponding to the duality of equivariant mixed Hodge modules, and prove a new reciprocity formula in terms of this duality. This unifies and generalizes the classical reciprocity formula of Brion-Vergne in Ehrhart theory as well as the above-mentioned more recent combinatorial formula of Beck-Gunnells-Materov.