Equivariant Ehrhart Theory

• Equivariant Ehrhart theory is the study of lattice point counts in polytopes under finite group actions, refining classical geometric and combinatorial methods.
• It employs equivariant generating functions that encode fixed-point data as character values, revealing rationality and quasipolynomial behavior in counting functions.
• The framework has applications in toric geometry, algebraic combinatorics, and topology, notably through invariant triangulations and equivariant Hilbert series computations.

Equivariant Ehrhart theory is the paper of lattice point enumeration in polytopes equipped with a finite group action, enriching classical Ehrhart theory with representation-theoretic and homological data. Central to this field are the equivariant generating functions, which record not merely the counts of lattice points in polytope dilates but their permutation representations under the group action. This framework unifies and extends geometric, combinatorial, and representation-theoretic perspectives on polytopal enumeration, with key implications for toric geometry, algebraic combinatorics, and topological invariants of manifolds.

1. Fundamental Definitions and Structures

Let $M\simeq \mathbb{Z}^d$ be a lattice, $P\subset\mathbb{R}^d$ a lattice polytope, and $G$ a finite group acting linearly or affinely on $M$ and preserving $P$. For each integer $k\geq 0$, $G$ acts on $kP\cap M$ by permutation. The equivariant Ehrhart theory replaces the scalar Ehrhart polynomial with a character-valued function: $\chi_{kP} \in R(G)$ where $R(G)$ is the complex representation ring of $G$, and $\chi_{kP}(g)$ is the number of fixed lattice points under $g$. The associated generating series, the equivariant Ehrhart series, is

$$E^G_P(t) = \sum_{k \geq 0} \chi_{kP}\, t^k \in R(G)[[t]].$$

This formal power series encodes for each group element $g$ the generating function for the fixed-point polytope $P_g := P \cap M^g$.

This setup is compatible with more general frameworks, such as Ehrhart theory over abelian group rings, where the character-valued sum is subsumed by the group ring of $G$ (Davis et al., 13 Nov 2025). In all cases, the classical Ehrhart series is recovered by specialization at the identity element of $G$.

2. Rationality, Quasipolynomiality, and Reciprocity

The structure of the equivariant Ehrhart series mirrors key properties from classical Ehrhart theory, with the following foundational results (Stapledon, 2010, Davis et al., 13 Nov 2025):

• Rationality: The series $E^G_P(t)$ is a rational function in $t$, typically of the form

$$E^G_P(t) = \frac{H^*_G(P; t)}{(1-t)\det(I - \rho\, t)}$$

where $\rho: G \to GL(M_{\mathbb{C}})$ is the induced representation on $M$, and $H^*_G(P; t)$ is the equivariant $h^*$-polynomial, with coefficients in $R(G)$.

• Quasipolynomiality: For each group element $g$, the function $k \mapsto \chi_{kP}(g)$ is a quasipolynomial in $k$ of degree $\dim P_g$, with period dividing the exponent of $G$ (Stapledon, 2010).
• Reciprocity: There is a direct equivariant analogue of Ehrhart–Macdonald reciprocity:

$(-1)^d\,\chi_{-kP} = \chi^*_{kP}\cdot\det(\rho)$

where $\chi^*_{kP}$ counts interior points (Stapledon, 2010).

• Denominator Structure: The denominator reflects the group action; for polytopes with an invariant triangulation, the denominator can be factored into terms $(1-t^{d_j})$ depending on the action, and the numerator $f(t)$ is, under translative or color-preserving actions, an effective sum of irreducible characters (D'Alì et al., 2023).

These results extend to multivariable, weighted, and abelian-group-valued variants, with rationality always guaranteed via equivariant Brion--Stanley theory (Davis et al., 13 Nov 2025, Maxim et al., 5 May 2024).

3. Invariant Triangulations and Hilbert Series

A central technical device is the equivariant version of the Betke–McMullen theorem: if $P$ admits a $G$-invariant unimodular triangulation $\Delta$, then the equivariant Ehrhart series coincides with the equivariant Hilbert series of the Stanley–Reisner ring $\mathbb{C}[\Delta]$ (D'Alì et al., 2023): $E^G_P(t)=\Hilb^G(\mathbb{C}[\Delta],t).$ When the action is translative (color-preserving in an appropriate coloring), one can choose a $G$-invariant linear system of parameters, and obtain

$\Hilb^G(\mathbb{C}[\Delta], t) = \frac{f(t)}{(1-t)^{\dim P + 1}}$

where $f(t)$ decomposes as a nonnegative sum of irreducible characters of $G$ (D'Alì et al., 2023, Clarke et al., 2022).

More generally, for any finite simplicial complex $\Sigma$ with a $G$-action by simplicial automorphisms, the equivariant Hilbert series is

$\Hilb^G(\mathbb{C}[\Sigma],t) = \frac{\sum_{i=0}^{\dim\Sigma+1} h_i^\Sigma t^i}{(1-t)^{\dim\Sigma+1}}$

with each $h_i^\Sigma \in R(G)$ effective if $\Sigma$ is Cohen–Macaulay and the action is translative. This yields explicit rational forms for many combinatorially and geometrically significant polytopes.

4. Key Examples and Combinatorial Models

Numerous families of polytopes admit full descriptions of their equivariant Ehrhart theory, often with explicit combinatorial models:

Alcoved Polytopes, Order Polytopes, and Lipschitz Poset Polytopes

For order polytopes $O(X)$ of a poset $X$, the automorphism group $G \subseteq \operatorname{Aut}(X)$ acts translatively, with the equivariant Ehrhart series realized as the Hilbert series of a regular unimodular triangulation indexed by chains of order ideals. Each coefficient in the numerator corresponds to a homology representation associated to flag complexes of the poset (D'Alì et al., 2023).

For Lipschitz poset polytopes, which are alcoved polytopes admitting unimodular triangulations preserved by the poset automorphism group, the equivariant Ehrhart series has a “flag-h” expansion paralleling the order polytope case. In some settings (e.g., star posets with symmetric group actions), explicit factorizations of the numerator reflect deep algebraic structure (D'Alì et al., 2023).

Hypersimplices and Decorated Ordered Set Partitions

For the hypersimplex $\Delta_{k,n}$, the symmetric group $S_n$ acts by coordinate permutation. Recent results show that the evaluation at $t=1$ of the equivariant $H^*$-polynomial is the permutation character on the set of hypersimplicial decorated ordered set partitions (DOSPs), providing a full combinatorial description of the equivariant Ehrhart theory of the hypersimplex (Clarke et al., 9 Dec 2024). This connects the theory to classical Eulerian numbers and supports permutation-character conjectures.

Permutahedra and Other Coxeter Polytopes

For the permutahedron $\Pi_n$ under the $S_n$-action, the equivariant Ehrhart theory involves explicit character formulas based on the description of fixed subpolytopes (zonotopes of lower dimension) and their lattice structure. Results include a full proof of the equivalence between effectiveness of the $H^*$-series and existence of invariant nondegenerate hypersurfaces when $n \leq 3$, and precise combinatorial criteria in higher rank (Ardila et al., 2019).

Rational and Pseudo-Integral Polytopes

The theory extends to rational polytopes and so-called pseudo-integral polytopes (PIPs), which exhibit polynomial counting functions despite irrational vertices. While the effectiveness of equivariant $H^*$-polynomial generally fails without integrality of vertices, the formalism for representation-valued Ehrhart series remains valid (Clarke et al., 2022, Davis et al., 13 Nov 2025).

5. Effectivity, Polynomiality, and Hypersurface Criteria

Key open problems concern the effectiveness of the equivariant $H^*$-polynomial (all coefficients honest representations), the criterion that it is a (finite) polynomial, and its interplay with equivariant geometry of toric hypersurfaces.

Stapledon's Effectiveness Conjecture

For a lattice polytope $P$ with $G$-action, the following are conjectured to be equivalent (Ardila et al., 2019, D'Alì et al., 2023):

1. The toric variety $X_P$ admits a $G$-invariant nondegenerate hypersurface.
2. The equivariant $H^*$-series is effective.
3. The equivariant $H^*$-series is a polynomial.

This conjecture is fully verified for the permutahedron with $n \leq 3$, and for graphic zonotopes and certain classes of hypersimplices, but both counterexamples and subtle failures appear for rational polytopes and in the case of the “trivial summand” conjecture for hypersimplices $\Delta_{2,n}$ (Clarke et al., 9 Dec 2024, Clarke et al., 2022).

Invariant Triangulations and Obstructions

When $P$ admits an invariant unimodular triangulation under $G$, the argument via Artinian reduction in the Stanley-Reisner ring produces effective and polynomial $H^*$. Conversely, the existence of torus-invariant nondegenerate hypersurfaces is obstructed by certain local combinatorial conditions (e.g., presence of odd rectangles in the face lattice) (Elia et al., 2022).

Weighted and K-theoretic Extensions

Recently, equivariant weighted Ehrhart theory has been developed, combining equivariant intersection cohomology, mixed Hodge modules, and generalized weightings on faces. The face-sum formulas for equivariant weighted Ehrhart polynomials reveal reciprocity and purity theorems at the level of equivariant Hodge polynomials, integrating toric geometry and motivic methods into the combinatorial framework (Maxim et al., 5 May 2024).

6. Techniques, Algorithms, and Representative Formulas

Multiple computational and conceptual techniques underlie modern equivariant Ehrhart theory:

• Symmetric Triangulation and Zonotopal Decomposition: Explicit half-open triangulations or zonotopal decompositions compatible with the group action allow calculation of the equivariant Ehrhart series and $H^*$-coefficients through orbit and forest enumeration (e.g., in graphic zonotopes) (Elia et al., 2022).
• Molien–McKay and Cartan Matrix Approaches: In the context of reflection or Weyl group actions (arising in Lie theory or Chern–Simons physics), equivariant Ehrhart series coincide with Molien sums enumerating group-invariant polynomials, often parameterized by the inverse Cartan matrix (Ju, 2023).
• Localization and Brion Theory: Techniques from algebraic geometry (e.g., Atiyah–Bott localization, equivariant Riemann–Roch) are deployed to establish polynomiality of weighted or group-valued Ehrhart series and to derive explicit character formulas (Maxim et al., 5 May 2024, Davis et al., 13 Nov 2025).
• Fixed Point and Fourier Methods: The behavior of the Ehrhart series under evaluation at group elements reduces to computations on fixed-point subpolytopes and is closely linked with harmonic analysis on finite groups (Fourier inversion, character theory) (Némethi et al., 2012).
• Algorithmic Implementations: Stepwise procedures for constructing $G$-invariant triangulations, computing character values on orbits, and extracting polynomial or homological invariants have been codified for computational platforms (e.g., SageMath) (Elia et al., 2022).

The following table summarizes some of the key structural formulas central to the theory:

Feature	Formula/Construction	Conditions/Context
Equivariant Ehrhart series	$E^G_P(t) = \sum_{k \geq 0} \chi_{kP}t^k$	Any finite group $G$
Rational form	$$E^G_P(t) = \frac{H^*_G(P;t)}{(1-t)\det(I - \rho\, t)}$$	Linear $G$-action, unimodular
Effectivity (under triangulation)	$H^*_G(P;t)$ effective (sum of true representations)	Translative action
Reciprocity	$(-1)^d\,\chi_{-kP} = \chi^*_{kP}\cdot\det(\rho)$	All $k > 0$, $d = \dim P$
Weighted generalization	$$E_{P,\varphi,f}(n;y) = \sum_{Q\preceq P}f_Q(y)(1+y)^{\dim Q + r}\sum\varphi(m)$$	$\varphi$ polynomial weight

7. Applications and Further Directions

Equivariant Ehrhart theory has direct implications in:

• Toric Geometry: Characterization of toric cohomology and connection to nondegenerate hypersurfaces, especially for identifying and constructing varieties with prescribed symmetry properties (Stapledon, 2010, D'Alì et al., 2023).
• Representation Theory: Equivariant Ehrhart polynomials realize representation-theoretic invariants (e.g., Weyl group actions, McKay correspondence), providing bridges with the cohomology of important moduli spaces and representation rings (Ju, 2023).
• Topology of 3-Manifolds: The coefficients of equivariant Ehrhart series for polytopes constructed from plumbing graphs encode invariants such as the Seiberg–Witten invariants for plumbed 3-manifolds (Némethi et al., 2012).
• Combinatorics: The theory informs enumeration in symmetric structures, such as descent statistics, Eulerian numbers, decorated partitions, and the fine structure of poset and graph-based polytopes (Clarke et al., 9 Dec 2024, Elia et al., 2022).

Future research addresses:

• Complete characterization of effectivity and polynomiality for large classes of polytopes and group actions.
• Extensions to weighted, multi-abelian, and motivic versions, integrating equivariant mixed Hodge structures (Maxim et al., 5 May 2024).
• Algorithmic development for high-dimensional and complex group actions.
• Deeper connections to mirror symmetry, moduli spaces, and arithmetic geometry, particularly through toric and motivic aspects (Davis et al., 13 Nov 2025).

Equivariant Ehrhart theory continues to serve as a critical junction of combinatorics, representation theory, and geometry, furnishing both conceptual insights and computational tools across mathematics.