Techniques in equivariant Ehrhart theory

Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including zonotopal decompositions, symmetric triangulations, combinatorial interpretation of the $h^\ast$-polynomial, and certificates for the (non)existence of invariant non-degenerate hypersurfaces. We apply these methods to several families of examples including hypersimplices, orbit polytopes, and graphic zonotopes, expanding the library of polytopes for which their equivariant Ehrhart theory is known.


Introduction
Ehrhart theory studies the enumeration of lattice points in polytopes via the Ehrhart counting function L(P ; t) := |tP ∩ M ′ | for positive integers t and a lattice M ′ .If P is a lattice polytope, then L(P ; t) agrees with a polynomial in t which is known as the Ehrhart polynomial [Ehr62].When a lattice polytope P is invariant under the linear action of a group G on M ′ with representation ρ : G → GL(M ′ ), then we may study its equivariant Ehrhart theory.This concept was introduced by Stapledon [Sta11] with motivation from toric geometry, representation theory, and mirror symmetry.The equivariant analogue of the Ehrhart polynomial is the character χ tP , defined as the character of the complex permutation representation on the lattice points in tP .The equivariant Ehrhart series is then given as follows: EE(P ; z) = t≥0 χ tP z t = H * (P ; z) det(I − z • ρ) Evaluating this series at the identity element of the group returns the usual Ehrhart series of P .The numerator H * (P ; z) is a formal power series with coefficients in R(G), the character ring of G.It is an open problem to determine for which polytopes and group actions H * (P ; z) is a polynomial and effective.Stapledon provided many conjectures around this topic, including: Conjecture 1.1 ([Sta11, Conjecture 12.1]).The following are equivalent: (i) The toric variety of P admits a G-invariant nondegenerate hypersurface with Newton polytope P .
(ii) The H * -series is effective.
(iii) The H * -series is a polynomial.
Stapledon further showed that (i) =⇒ (ii) =⇒ (iii); see Sections 2.4 and 2.5 for details.In email correspondence that they shared with the authors in 2019, Santos and Stapledon presented a counterexample to (iii) =⇒ (i), which we include here with their permission.
Proof.By computing the Ehrhart series of the rational fixed polytope of the cube under each of these rotations, and then comparing these to a character table for G, one can find the H * -series of P under this action to be χ triv + χ reg z + χ triv z 2 where these characters indicate the trivial and regular representations of G.So the H * -series is a polynomial and is effective.However, the toric variety of P does not admit a G-invariant nondegenerate hypersurface.Figure 1 shows the two orbits of the vertices of P under the action of G.So a G-invariant hypersurface in the toric variety of P has the equation 0 = (1 + z + xy + xyz) + c(x + y + xz + yz) = (1 + z)(1 + xy + c(x + y)) for some complex parameter c.Regardless of the value of c, such a hypersurface is singular when z = −1 and 1 + xy + c(x + y) = 0.
In this article we present a variety of techniques for computing and studying the equivariant Ehrhart series.Our aim is to develop equivariant Ehrhart theory as its own branch of discrete geometry, and to investigate Stapledon's conjectures for a number of families of polytopes.We demonstrate how to use several tools to describe the equivariant Ehrhart series, including zonotopal decompositions, symmetric triangulations, combinatorial interpretations of the h * -polynomial, and computational methods using Sagemath.These techniques serve as a guidebook for future progress toward resolving the many open questions and conjectures in this field.
We first provide more details about our setup in Section 2, as well as presenting the necessary background in equivariant Ehrhart theory and representation theory.We then arrive to our catalog of techniques in Section 3.
Section 3.1 investigates the case when P is a graphic zonotope and G is the automorphism group of the graph.This section generalizes past work on the permutahedron [ASV19, ASVM20], and serves as a blueprint for studying the equivariant Ehrhart theory of general zonotopes.The main results of this section are Corollary 3.14 which describes the equivariant Ehrhart series of a graphic zonotope, and Theorem 3.15 in which we characterize the conditions under which the H * -series of a graphic zonotope is a polynomial.
In Section 3.2, we use symmetric triangulations to calculate the equivariant Ehrhart series.The symmetric triangulations in question are called G-invariant half-open decompositions and are closely related to partitionability of posets.In Theorem 3.27, we generalize a theorem of Stapledon for the equivariant Ehrhart series of a simplex to lattice polytopes with G-invariant half-open decompositions.In Theorem 3.29, we show that χ tP is polynomial in t for polytopes with G-invariant half-open decompositions that satisfy an additional property on the group action, and give a formula for the equivariant Ehrhart series.This section appears in [Eli22].
Section 3.3 deals with instances where we can break down the H * -series using a combinatorial interpretation of its coefficients.In particular, we study hypersimplices under the action of the cyclic group, as well as permutahedra in prime dimensions.Theorem 3.33 uses decorated ordered set partitions to describe the H * -series of the hypersimplex, building off of [Kim20].In Theorem 3.47, we give an explicit formula for the H * -series of a polytope under a cyclic group action with trivial fixed subpolytopes.This is specified to the case of prime permutahedra in Corollary 3.48, which also shows the H * -series is polynomial and effective.
In Section 3.4 we focus on criteria (i) from Conjecture 1.1, demonstrating techniques for proving the existence or non-existence of G-invariant nondegenerate hypersurfaces.The main results of this section are Theorem 3.53 and Corollary 3.56 which characterize a family of S n -orbit polytopes for which no such hypersurface can exist, and Theorem 3.57 regarding hypersimplices, another subfamily of orbit polytopes which do exhibit these hypersurfaces.Finally, in Section 4, we collect the many open questions that arose during our work on this project, and in Appendix A, we demonstrate the calculation of the equivariant Ehrhart series using our implemented code which is staged for release with Sagemath version 9.6.

Background
We give relevant background on Ehrhart theory in Section 2.1.In Section 2.2, we present the general setup for equivariant Ehrhart theory that is used throughout.Section 2.3 presents background on representation theory, and Section 2.4 presents background on the equivariant Ehrhart series, in particular providing rational generating functions.In Section 2.5, we discuss G-invariant nondegenerate hypersurfaces and a useful corollary for their detection.In Section 2.6, we present some motivating lemmas on equivariant Ehrhart theory for restricted representations.

Ehrhart Theory
The main reference we follow for Ehrhart theory is [BR15].Fix a lattice M ⊂ R n .The Ehrhart counting function of a polytope P ⊂ R n , written L(P ; t), gives the number of lattice points in the t-th dilate of P for t ∈ Z ≥1 : L(P ; t) = |tP ∩ M |.
Ehrhart's theorem [Ehr62] says that if P is a lattice polytope (i.e. the vertices of P are contained in M ), then for positive integers, L(P ; t) agrees with a polynomial in t of degree equal to the dimension of P .Furthermore, the constant term of this polynomial is equal to 1 and the coefficient of the leading term is equal to the Euclidean volume of P within its affine span.The interpretation of other coefficients of the Ehrhart polynomial is an active direction of research, see for example [FJS21].
Generating functions play a central role in Ehrhart theory.The Ehrhart series Ehr(P ; z) of a polytope P is the formal power series given by Ehr(P ; z) = 1 + t≥1 L(P ; t)z t .
Let Cone(P ) := {x ∈ R n+1 : x = λ(y, 1), where y ∈ P, λ ≥ 0} as in Figure 2. We may view the coefficient of z t in the Ehrhart series as counting the number of lattice points in Cone(P ) at height t.This viewpoint allows us to show that if P is a d-dimensional lattice polytope, then the Ehrhart series has the rational generating function where h * (P ; z) = d i=0 h * i z i is a polynomial in z of degree at most d, called the h * -polynomial.Furthermore, each h * i is a non-negative integer [Sta80].The coefficients of the h * -polynomial form the h * -vector : (h * 0 , h * 1 , . . ., h * d ).The Ehrhart polynomial may be recovered from the h * -vector through the transformation Let P ⊆ R d be a rational d-polytope with denominator k, i.e., k is the smallest positive integer such that kP is a lattice polytope.Then L(P ; t) is a quasipolynomial with period dividing k, i.e., of the form L(P ; t) = c d (t)t d + • • • + c 1 (t)t + c 0 (t) where c 0 (t), c 1 (t), . . ., c d (t) are periodic functions.In this case, the Ehrhart series has the rational generating function Ehr(P ; z) := where h * (P ; z) ∈ Z[z] has degree < k(d + 1).The choice of denominator is no longer canonical.

The Equivariant Setup
For a lattice and Z-module M , we write M R for M ⊗ Z R.
Setup 2.1.Let G be a finite group acting linearly on a lattice M ′ ∼ = M × Z of rank d + 1 such that the Z-coordinate of the lattice points in M ′ is preserved under the action of G. Let P ⊂ M ′ R be a d-dimensional G-invariant polytope with vertices in M × {1}, where G-invariant means that as a set, g(P ) = P , for all g ∈ G.
We assume M = Z d when convenient.The exponent of a finite group G is the smallest positive integer N such that g N = id G for all g ∈ G.For an element g ∈ G, the subset of P fixed by g, denoted P g and called the fixed subpolytope, is the convex hull of the barycenters of orbits of vertices of P under the action of g [Sta11,Lemma 5.4].This implies that any fixed subpolytope P g for g ∈ G is a rational polytope with denominator dividing N .

Representation Theory
For a nice introduction to representation theory of finite groups, see [Sag01] or [Ser77].Let V be an n-dimensional vector space over a field K of characteristic 0. A representation ρ of a group G on V is a group homomorphism ρ : G → GL(V ) from G to the group of invertible K-linear transformations of V .Equivalently ρ may be seen as a group homomorphism from G to the group GL n (V ) of n × n invertible matrices with entries in K.A subspace W ⊆ V is called G-invariant if W = g(W ) as a set, for all g ∈ G.A representation is irreducible if there are no nontrivial, proper invariant subspaces of V under the action of G.
Let ρ : G → GL n (V ) be a representation of G.The character of ρ, written χ ρ , is the function G → C such that χ ρ (g) := trace(ρ(g)); the trace of a matrix is the sum of its diagonal entries.The characters of irreducible representations are referred to as irreducible characters.A group G always has a trivial representation on a vector space V which sends each group element to the identity map on V .The character of the trivial representation is denoted by χ triv throughout.Characters are class functions, functions from the group to the complex numbers that take the same value on every conjugacy class.In fact, a class function is a character of a representation if and only if it can be written as a nonnegative integral linear combination of the irreducible characters of G.We write R + (G) for the set of the class functions that are characters and refer to them as effective characters.The group generated by R + (G) is denoted R(G) and called the character ring; it consists of all class functions that can be written as linear integral combinations of the irreducible characters.Elements of R(G) are referred to as virtual characters.The character ring R(G) is a subring of the C-vector space F C (G) of class functions on G with values in C, called the ring of class functions.There is an inner product on F C (G) such that for two class functions φ, χ of G, φ, χ = 1 |G| g∈G φ(g)χ(g), where • denotes the complex conjugate.Let a group G act on a finite (ordered) set S. The permutation representation of G on S is the group homomorphism that sends every element of G to the corresponding permutation matrix for the action of G on S. The character of the permutation representation is called the permutation character.

Equivariant Ehrhart Theory
Let χ tP be the complex permutation character induced by the action of G on the lattice points in tP ∩M ′ .Definition 2.2.The equivariant Ehrhart series, EE(P ; z), is the formal power series in R(G) [[z]], such that the coefficient of z t for t ∈ Z ≥0 is the character χ tP : Evaluating the series at g ∈ G, EE(P ; z)(g), gives the Ehrhart series of the fixed subpolytope P g [Sta11, Lemma 5.2].
Theorem 2.3 ([Sta11, Theorem 5.7]).Let P be a lattice d-polytope invariant under the action of a group G as in the Setup 2.1 and exponent N ≥ 1.As a function of t, χ tP is quasipolynomial, that is, where f 0 (t), f 1 (t), . . ., f d (t) ∈ F C (G) are periodic functions in t with period dividing N .
Proof.For all g ∈ G, the Ehrhart counting function L(P g ; z) can be expressed as a quasipolynomial of period N and degree equal to dim(P g ) ≤ d.Therefore, where c g j,i ∈ Q is the coefficient of the degree i term in the j-th constituent of the Ehrhart quasipolynomial of P g .Let {g 1 , . . ., g k } be conjugacy class representatives of G. Define class functions f j,i [g ℓ ] := c g ℓ j,i for all j ∈ [N ], ℓ ∈ [k], and i ∈ {0, 1, . . ., d}.Then for j ∈ [N ], and t ≡ j mod N , . For all t ≡ j mod N , χ tP may be expressed as a linear combination of the irreducible characters of G with coefficients in Q[t].
Corollary 2.5.Let P be a rational polytope with denominator S ∈ Z >0 such that P is invariant under the linear action of a group G. Then χ tP is quasipolynomial in t with period dividing SN .
Theorem 2.6.Let P be a G-invariant lattice polytope of dimension d as in the Setup 2.1, where G has exponent N and irreducible characters {χ 1 , . . ., χ k }.The equivariant Ehrhart series has the following rational expression: where We therefore have two different rational expressions for the equivariant Ehrhart series: , Remark 2.7.In [Sta11], the H * -series is denoted by φ, and ρ is the representation of G acting on M R .Accordingly, the denominator in [Sta11] is Understanding the H * -series is the main goal of Stapledon's Conjecture 1.1, and he stated that (ii) implies (iii).We provide a short proof: Lemma 2.8.If H * (P ; z) is effective then H * (P ; z) is a polynomial.
Proof.The H * -series is a priori an infinite formal sum: H * (P ; z) = i≥0 H * i z i .On the level of series, we have: with c i,j ∈ Z ≥0 for all i, j.As χ j (id G ) > 0 for all j, no cancellation can occur and H * (P ; z) must be polynomial.

Nondegenerate Hypersurfaces
One motivation for considering the H * -series is its connection to toric geometry, see [Sta11,Sections 7 and 8].A lattice polytope P ⊂ R d defines a projective toric variety X P .A hypersurface in X P is given by the vanishing set of f = v∈P ∩Z d c v x v , where c v ∈ C and x v denotes the monomial x v1 1 x v2 2 • • • x vn n .The Newton polytope of f is the convex hull of the lattice points corresponding to monomials of f that have nonzero coefficients c v .A hypersurface has Newton polytope P if c v = 0 for all vertices v of P .Suppose that a lattice polytope P is invariant under the linear action of a group G. Then a hypersurface in X P is said to be G-invariant if c v = c w for all lattice points v, w in the same G-orbit of P ∩ Z d .The hypersurface is smooth if the gradient vector ( ∂f ∂x1 , . . ., ∂f ∂xn ) is never zero when x 1 , . . ., Theorem 2.9 ([Sta11, Theorem 7.7]).If there exists a G-invariant nondegenerate hypersurface with Newton polytope P , then H * (P ; z) is effective.In particular, this assumption holds if the linear system Γ(X P , L) G of G-invariant global sections on the corresponding line bundle L is base point free.
Recall that a subspace W ⊆ Γ(X, L) is basepoint free if for every p ∈ X, there exists s ∈ W with s(p) = 0 [CLS11, Section 6.0].For a face Q ⊂ P let G Q denote the stabilizer of Q.By [Sta11, Remark 7.8], the linear system of G-invariant global sections Γ(X P , L) G on X P is basepoint free if and only if for each face Q ⊂ P , the linear system on the torus T is basepoint free.Additionally, this condition is automatically satisfied for faces of dimension ≤ 1. Suppose that every face To show that the linear system (3) is basepoint free on T , we need to show that for every x ∈ T , there exists a polynomial in the system that doesn't vanish at x.In fact, we have something stronger, since the polynomial x uQ is an element of the linear system and doesn't vanish at any x ∈ T .Thus we recover the following combinatorial criterion that certifies the existence of a G-invariant nondegenerate hypersurface with Newton polytope P .
Corollary 2.10 ([Sta11, Corollary 7.10], reworded).If every face Q of P with dim(Q) > 1 contains a lattice point that is G Q -fixed, where G Q denotes the stabilizer of Q, then the toric variety of P admits a G-invariant nondegenerate hypersurface with Newton polytope P .

Equivariant Ehrhart Theory of Restricted Representations
Let P be a polytope invariant under the action of G as in the Setup 2.1, and let H be a subgroup of G.The action of G on P induces an action of H on P by restriction.Let H * G (P ; z) and H * H (P ; z) be their respective equivariant H * -series.
Theorem 2.11.If the toric variety of P admits a G-invariant nondegenerate hypersurface with Newton polytope P , then the same hypersurface is a nondegenerate H-invariant hypersurface.
Proof.Let v∈P ∩Z d c v x v = 0 be a G-invariant nondegenerate hypersurface.Every H-orbit of lattice points in P is contained in a G-orbit of lattice points.Thus c v = c w for all v, w in an orbit of H, and the hypersurface is H-invariant.The non-degeneracy condition is independent of the group.
Theorem 2.12.If H * G (P ; z) is effective then H * H (P ; z) is effective.
Proof.The coefficient of each z i in H * G (P ; z) is an effective character.Thus for each coefficient there is a corresponding representation of G on a vector space which is a direct sum of irreducible representations of G. Restricting to the action of H decomposes each summand further into a finite number of irreducible representations of H. Theorem 2.13.If H * G (P ; z) is polynomial, then H * H (P ; z) is polynomial.
Proof.Let {χ 1 , . . ., χ m } be the irreducible representations of G, and let {µ 1 , . . ., µ k } be the irreducible representations of H. Suppose H * G (P ; z) is polynomial so that where c i,j ∈ Z for all i ∈ {0, . . ., d} and j ∈ We compute χ i | H , µ j H for all pairs i, j in order to restrict H * S4 (Π 4 ; z) to H * H (Π 4 ; z).A factor of 1 + z appears in the numerator:

Techniques
In this section we introduce our main results, exploring four approaches toward equivariant Ehrhart theory and applying these methods to a variety of examples.

Zonotopal Decompositions
Zonotopes are a family of polytopes which can be decomposed into parallelotopes, and from these decompositions their Ehrhart polynomials are easily computed.This approach also lends itself nicely to equivariant Ehrhart theory, and has previously been used to study the equivariant Ehrhart theory of the permutahedron under the action of the symmetric group [ASV19,ASVM20].Here we adapt this technique to all zonotopes of the Type A root system, also known as graphic zonotopes.We expect this to generalize further to other root systems (see [ABM20] for progress in this direction), as well as to general  Let Γ = (V, E) be a simple graph, where V is a finite set of vertices, and E is the set of undirected edges.We write E ⊆ V 2 where V 2 is the collection of all 2-element subsets of V , and we denote by RV a real vector space with a basis {e v : v ∈ V } indexed by the elements of V .The graphic zonotope Z Γ is given by taking the Minkowski sum of line segments in RV The dimension of Z Γ is |V | minus the number of connected components of Γ.The corresponding graphic zonotope is 1≤i<j≤n [e i , e j ], also known as the permutahedron.
The following characterizations of the graphic zonotope are well-known; see [Sta07, Section 2] and [AA17, Section 13] for more information.

• ...the half-space description
where for any subset S ⊆ V , we define inc Γ (S) to be the number of edges of Γ that have an endpoint in S (are "incident to S").

• ...the vertex description
where indeg o (v) is the number of edges pointing into the vertex v in the acyclic orientation o of Γ.Furthermore, v o is a vertex of Z Γ for all acyclic orientations o of Γ.
The symmetric group Sym(V ) acts on the set of vertices V , also inducing an action on V 2 .The automorphism group Aut(Γ) of Γ is the subgroup of Sym(V ) under which the edge set E is invariant.Just as Aut(Γ) acts on V and holds Γ invariant, likewise Aut(Γ) acts linearly on RV and holds Z Γ invariant.Hence RV is a representation of Aut(Γ), and we can consider the equivariant Ehrhart theory of Z Γ under the action of Aut(Γ) (or any subgroup thereof).
For the setup, we will view Aut(Γ) as a subgroup of Sym(V ), so every element σ ∈ Aut(Γ) can be written as a product of m ≤ |V | disjoint cycles σ 1 , . . ., σ m partitioning V .We denote the cycle lengths by ℓ 1 , . . ., ℓ m , respectively.For each σ ∈ Aut(Γ), we define the σ-connectivity graph C Γ (σ) to be the simple graph with vertex set {σ 1 , . . ., σ m } and an edge between two distinct cycles σ i and σ j whenever some element of the cycle σ i is connected by an edge in Γ to some element of σ j .Lemma 3.6.Let σ ∈ Aut(Γ) and let σ i , σ j be (not necessarily distinct) cycles of σ.Then for all v ∈ σ i , the number of edges connecting v to the elements of σ j is constant.
Proof.Suppose that u, v ∈ σ i have different numbers of edges connecting them to σ j .Since u and v are in the same cycle of σ, there exists r such that σ r (u) = v.But σ r is then not an automorphism of Γ, a contradiction.
We denote the number described in Lemma 3.6 by deg(σ i , σ j ).(Warning: It is NOT true in general that deg(σ i , σ j ) = deg(σ j , σ i ); see Example 3.7.)We will repeatedly make use of the following identity for distinct cycles σ i and σ j : We also write the number of edges within a given cycle σ i as E ii and note that Observe also that the notation E ij is symmetric in i and j.
We use e σi to denote the vector v∈σi e v ∈ RV .Theorem 3.8 and its proof closely follow the description of the fixed polytopes of the permutahedron from Theorem 2.12 of [ASV19].Our result generalizes theirs: where they used permutations, we use acyclic orientations of Γ, and we also make use of identity (4).
Theorem 3.8 (Description of graphic zonotope fixed polytopes).Let σ ∈ Aut(Γ).Then the following descriptions of the fixed polytope Z σ Γ are equivalent: (i) It is the subset of Z Γ that is fixed by the action of σ.
(ii) It is the convex hull of the points w o , where o ranges over acyclic orientations of C Γ (σ), and Furthermore, this gives a bijection between acyclic orientations of C Γ (σ) and vertices of Proof.(ii) ⊆ (i): Let ô be any orientation of Γ "extending" o, meaning that if σ i → σ j in o, and v ∈ σ i and v ′ ∈ σ j are joined by an edge of Γ, then v → v ′ in ô.Then w o is the average of the σ-orbit of the vertex of Z Γ corresponding to ô, so w o ∈ Z Γ .By definition, the coordinates of w o are constant over the cycles of σ, so w o ∈ Z σ Γ for all acyclic orientations o.
(iii) ⊆ (ii): Let w be a vertex of (5) and let y = (y 1 , . . ., y n ) ∈ (R n ) * be such that w is the y-maximal face of (5).For 1 ≤ i ≤ m, define y σi to be the average of y k for k ∈ σ i ; so y σi = 1 ℓi k∈σi y k .Now, since y is maximal at a vertex, we know that for each edge {σ i , σ j } of C Γ (σ), we must have Using identity (4), we find that y σj = y σi .For each edge {σ i , σ j } in C Γ (σ), if y σi < y σj , assign the direction pointing from σ i to σ j .This gives an acyclic orientation o of C Γ (σ), and w = w o .
(i) ⊆ (iii): According to [Sta11, Lemma 5.4], the fixed polytope Z σ Γ is the convex hull of the averages over the σ-orbits of the vertices of Z Γ .So we need only show that for any vertex v o of Z Γ corresponding to the acyclic orientation o of Γ, the average v o of the σ-orbit of v o is in the Minkowski sum (5).To do this, we will fix an orientation o ′ of Γ, show that v o ′ is in the Minkowski sum, and then show that we can find a path from v o ′ to v o within the sum.
We construct o ′ as follows: Choose any ordering of the n vertices of Γ such that all the vertices in the cycle σ i are less than all the vertices in σ i+1 for 1 ≤ i < m, and let o ′ be the acyclic orientation of Γ induced by this ordering of the vertices.Then the vertex v o ′ of Z Γ is defined as in Proposition 3.4, and the average of the σ-orbit of which is clearly seen to be in the sum (5).Now consider any acyclic orientation o of Γ.Let o 0 := o ′ , o 1 , . . ., o ℓ := o be a sequence of acyclic orientations of Γ where o k is obtained from o k−1 by reversing the orientation of one edge for 1 ≤ k ≤ ℓ.For each k, if the edge of o k−1 that is reversed to obtain o k points from a vertex in σ i to a vertex in σ j , then we have that Note that if the edge that is reversed connects two vertices of Γ that are in the same cycle of σ, then there is no change in the σ-average.Now for each pair of distinct cycles σ i and σ j , let D ij be the number of edges between σ i and σ j that have different orientations in o ′ and o.Then we get If there are no edges between σ i and σ j in Γ, then D ij must be 0, so deg(σ i , σ j )e σi − deg(σ j , σ i )e σj may only contribute to this sum if there is an edge between σ i and σ j in C Γ (σ).Furthermore, the maximum possible number of edges between σ i and σ j is E ij , so 0 ≤ Dij Eij ≤ 1. Hence adding our earlier expression for v o ′ produces a point in the zonotope (5).
Finally, it remains to prove that the acyclic orientations of C Γ (σ) are in bijection with the vertices of Z σ Γ .By definition, each acyclic orientation o results in a distinct point w o .We already showed that every vertex of Z σ Γ has the form w o for some acyclic orientation o.For any w o , let y = m i=1 y i ( k∈σi e * k ) be some linear functional in (R n ) * such that y i < y j whenever σ i → σ j in o.Then y(deg(σ j , σ i )e σj ) = E ij y j , so we can see that exactly one endpoint of each of the line segments comprising (5) is maximized by y, and the resulting sum gives exactly w o .So w o is the unique point in Z σ Γ maximizing y, and is hence a vertex.
In general, a zonotope generated by line segments with direction vectors s 1 , . . ., s k can be tiled by half-open parallelotopes that are in bijection with linearly independent subsets of {s 1 , . . ., s k }.In the case of Z σ Γ , the generating vectors are We define a subforest of C Γ (σ) to be any graph on the same vertex set {σ 1 , . . ., σ m } as C Γ (σ) whose edge set is a subset of the edges of C Γ (σ), and that does not contain any cycles.
Proposition 3.9.The linearly independent subsets of the vectors in (6) are in bijection with subforests of the σ-connectivity graph C Γ (σ).
Proof.The bijection is as follows: Given a subgraph of C Γ (σ), for each edge {i, j} where i < j, take the vector deg(σ i , σ j )e σi − deg(σ j , σ i )e σj to be in the subset; similarly, given a subset of the generating set, take the corresponding edges of C Γ (σ).It remains to show that linearly independent subsets of the generators correspond exactly to subforests of C Γ (σ).By identity (4), the generating vector corresponding to the edge {i, j} is simply a positive multiple of the vector From here we can follow exactly the proof of [ASV19, Lemma 3.2], noting that our generators are a subset of theirs and therefore inherit the same linear independence properties.
For a given subforest F of C Γ (σ), we denote by F the corresponding half-open parallelotope in the tiling of Z σ Γ .
Proposition 3.10.The volume of F is given by the formula Proof.This follows directly from [ASV19, Lemma 3.3].
The 2-valuation of a positive integer k, denoted val 2 (k), is the highest power of 2 dividing k.For example, val 2 (24) = 3.
Proof.The affine span of F is the span of the vectors deg(σ i , σ j )e σi − deg(σ j , σ i )e σj corresponding to to the edges {σ i , σ j } of F , translated by the vector Let y = m i=1 y i e σi be a point in aff( F ). Then the coordinates y i satisfy, for all connected components T of F , σj ∈T We want to determine when (9) has integer solutions y.It is a fact from elementary number theory that this occurs exactly when gcd(ℓ j : σ j ∈ T ) divides the right side of (9).Clearly each E ij is divisible by gcd(ℓ j : σ j ∈ T ), so we may focus our attention on the term σj ∈T E jj .If we multiply this term by 2, then we get σj ∈T ℓ j deg(σ j , σ j ) which is clearly divisible by gcd(ℓ j : σ j ∈ T ).Hence we need only consider 2-valuations to determine whether divisibility holds.Finally, noticing that val 2 (gcd(ℓ j : σ j ∈ T )) = min σj ∈T val 2 (ℓ j ) allows us to conclude that the condition of (σ, Γ)-compatibility introduced in Definition 3.11 is exactly what we need to guarantee integer solutions.
Let c(F ) be the number of connected components of a graph F , and let m be the number of cycles of σ ∈ Aut(Γ).Note that for a subforest F of C Γ (σ), the number of edges of F is m − c(F ).We omit proofs for the following two corollaries as they mirror the proofs of Theorem 1.1 and Proposition 5.1 in [ASVM20].
Corollary 3.13.The Ehrhart quasipolynomial of the fixed polytope Eulerian polynomials commonly arise when computing the Ehrhart series of polytopes.The Eulerian polynomial A n (z) is defined by the identity Notice that A n (z) has no constant term; the coefficients of 1 z A n (z) are called the Eulerian numbers.We can describe them using ascents of permutations, where the position i ∈ {1, 2, . . ., n − 1} is an ascent is the number of permutations in S n with exactly k ascents, denoted A n,k .
Corollary 3.14.The Ehrhart series of the fixed polytope Z σ Γ has the following form: To assess whether H * (Z Γ ; z) is a polynomial, we need to examine for which σ ∈ Aut(Γ) the poles of Ehr(Z σ Γ ; z) cancel with the zeros of det( , where m is the number of cycles of σ and ℓ 1 , . . ., ℓ m are the cycle lengths.The series Ehr(Z σ Γ ; z) always has a pole at z = 1, may have a pole at z = −1, and never has a pole at any other value of z.We can see from Corollary 3.14 that the pole at z = 1 has order at most m, and since m i=1 (1 − z ℓi ) has a zero of order m at z = 1, the pole at z = 1 will always cancel.Hence we may focus on the pole at z = −1.The following theorem tells us when this pole cancels.(c) There exists an even ℓ i with deg(σ i , σ i ) odd, and Proof.In case (a), all subforests of C Γ (σ) are (σ, Γ)-compatible since val 2 (ℓ i ) = 0 for all i.So the Ehrhart series of Z σ Γ does not have a pole at z = −1.Likewise in case (b); here we can also see that the condition of (σ, Γ)-compatibility from Definition 3.11 is always satisfied.Now assume (a) and (b) both fail.Then we must have some cycle σ i with even length ℓ i where deg(σ i , σ i ) is odd.In this case, we are guaranteed to have at least one (σ, Γ)-incompatible subforest F of C Γ (σ), since every forest where σ i is alone in a connected component will be (σ, Γ)-incompatible.From Corollary 3.14 we see that the pole of Ehr(Z σ Γ ; z) at z = −1 has order max (σ,Γ)-incomp.F {m − c(F ) + 1}, which is one more than the maximum number of edges in a (σ, Γ)-incompatible subforest of C Γ (σ).Since We conclude this section by focusing in on a particularly simple family of graphs.Consider the path graph Path n as introduced in Example 3.2.Then Aut(Path n ) = Z/2Z for all n, but the cycle type of the group generator depends on whether n is even or odd.When n is even, the generator of Aut(Path n ) written in cycle notation is σ = (1 n)(2 n − 1) . . .( n 2 n 2 + 1), and when n is odd, the generator is 2 ).The corresponding connectivity graphs C Path n (σ) are shown in Figure 5.
We first consider the case when n is even.Let F be any subforest of C Path n (σ).Using formula (7), we get We next assess (σ, Path n )-compatibility of F .Since ℓ i = 2 for all cycles σ i , the left side of inequality (8) will always be 1.However, whenever T is the connected component of F containing the cycle ( n 2 n 2 + 1), the right side of inequality (8) will be 0 since this particular cycle contains one edge of Path n and all other cycles have no edges within them.This means that no subforest Hence when n is even, the Ehrhart quasipolynomial of Z σ Pathn is 0 for odd dilations and (t + 1) n/2−1 (the Ehrhart polynomial of an integral parallelotope) for even dilations t: Next, using Corollary 3.14 we get and we can see directly from this Ehrhart series that H * (Z Pathn ; z) is a polynomial when n is even.Now suppose n is odd.As in the even case, we can use (7) to find that vol F is again 1 for all subforests F of C Path n (σ).However, unlike the even case, we have deg(σ j , σ j ) = 0 for all cycles σ j of σ.This means that the half-integral shift of equation ( 5) disappears in this case, and we are left with an integral polytope Z σ Pathn .Since all the fixed polytopes are integral in this case, the H * -series is guaranteed to be a polynomial.
Let us go further and show that the H * -polynomial is effective when n is odd.The fixed polytopes Z id Pathn and Z σ Pathn are both lattice parallelotopes with relative volume 1 and dimensions n − 1 and n−1 2 , respectively.So we get where we have arranged both series to have the desired denominator det(I − ρ • z).Notice that the numerators of both rational functions are polynomials of degree n − 2 with positive integer coefficients, and they are H * (Z Path n ; z) evaluated at the elements of Plugging in id and σ to H * (Z Path n ; z) gives Summing these two equations, it is clear to see that a k > 0, which supports [Sta11,Conjecture 12.4].Subtracting the second equation from the first, we see that b k ≥ 0 if and only if the right side of the first equation is greater than or equal to the right side of the second equation.This is indeed true, and we prove it combinatorially.
, the number of permutations of the set {1, 2, . . ., n − 1} with exactly k ascents.On the other hand, we have To see that (10) ≤ A n−1,k , we observe that (10) counts some, but not all, of the permutations in S n−1 with exactly k ascents.The first summand of (10) deals with the case where the elements 1, 2, . . ., n−1 2 come before the remaining elements in the permutation.The Eulerian number A (n−1)/2,i counts the number of permutations of {1, 2, . . ., n−1 2 } with exactly i ascents; we are then guaranteed an ascent at position n−1 2 , and then the binomial coefficient gives the number of ways to choose the positions of the remaining k − i − 1 ascents from {(n + 1)/2, (n + 3)/2, . . ., n − 2} (which is less than or equal to the number of ways to complete the permutation with ascents in those positions).Likewise, the second summand of (10) deals with the case where the elements n+1 2 , n+3 2 , . . ., n − 1 come before the remaining elements in the permutation; in this case we are guaranteed to have no ascent at position n−1 2 .Using this reasoning, it is clear that (10) is less than or equal to the number of permutations in S n−1 with exactly k ascents where the elements {1, 2, . . ., n−1 2 } occur either in the first n−1 2 positions or in the last n−1 2 positions.This in turn is clearly less than or equal to A n−1,k .Since a k , b k ≥ 0 for all k, the H * -polynomial of Z Path n is effective with respect to the action of Z/2Z when n is odd.

Invariant Half-Open Decompositions
In this section, we use G-invariant triangulations to calculate the equivariant Ehrhart series.More precisely, the triangulations are half-open decompositions.For background definitions in discrete geometry see [Zie95].
Definition 3.18.A half-open decomposition of a lattice polytope P with lattice triangulation T is a partition of the face poset of T into intervals such that the empty set is not in its own class.We use T [ ) to refer to the half-open decomposition, i.e., the triangulation together with the partition of the face poset.We identify a simplex S in T with its set of vertices when convenient.
Definition 3.18 is closely related to partitionability; the face poset of a pure simplicial complex is partitionable if it can be partitioned into disjoint intervals such that each maximal element is a facet of the complex, see [Sta96,Chapter 2].In this case, it is easy to read off the h-vector of the simplicial complex by recording the heights of the minimal simplices in the intervals.Definition 3.18 generalizes the typical method for creating a half-open decomposition of (the cone over) a triangulated polytope by choosing a generic vector in the interior of one simplex and then taking "sunny-side" of each face ; one asks if it is possible to remain in a facet of the triangulation when walking in the direction of the vector , see [BS18, Secton 5.3] or [HNP21, Section 3.4] for details.Using a generic vector in the typical method creates a shelling (and more weakly a partition) of the simplicial complex.However, this typical method can fail to create symmetric half-open decompositions with respect to the group action, and does not easily allow for simplices of varying dimension in the triangulation, highlighting the usefulness of the more general half-open decompositions from Definition 3.18.
This implies that all vertices of S have a coefficient in R >0 , and all vertices of v ∈ S \ S have a coefficient in R ≥0 .Thus, z ∈ Cone(I).It remains to show that the union I∈T [ ) Cone(I) is disjoint.Suppose z = 0, z ∈ Cone(I), and z ∈ Cone(I ′ ) for two intervals I = [S, S], for a unique face T of the simplicial complex.As z is in Cone(I), where λ i ∈ R >0 and µ j ∈ R ≥0 .Restricting to summands with strictly positive coefficients expresses z as a conical combination of vertices of T , since it is an expression of the form (11).This also shows that S ⊂ T .Likewise, S ′ ⊆ T ⊆ S ′ .Thus T is contained in two intervals in the half-open decomposition, and they must be the same.We present a lemma on the representation theory of the symmetric algebra, which is useful in Theorem 3.27, to follow.Through it, we can see one motivation for considering the denominator det(I − z • ρ) to express the equivariant Ehrhart series.Let {e 1 , . . ., e n } be a basis for a vector space V .The set e i 1 ⊙ • • • ⊙ e i t | i ℓ ≤ i ℓ+1 is a basis for Sym t (V ), the t-th symmetric power.A basis for the t-th exterior power, Λ t (V ), is the set The group G acts diagonally on Sym t (V ) and Λ t (V ) in the same manner.It is useful to look at the characters of these actions, which we denote by χ Sym t (V ) and χ Λ t (V ) respectively.For g ∈ G, let {e 1 , . . ., e n } be an orthonormal eigenbasis of the action of g on V , with eigenvalues {λ 1 , . . ., λ n }.This is possible as ρ(g) can be written as a unitary matrix, see for example [Ser77, Chapter 1].Then where This shows that an eigenvector of Sym t (V ) has an eigenvalue that is a monomial in the eigenvalues of ρ(g) of degree t.So the sum of the eigenvalues of g acting on Sym t (V ) is the sum of all monomials in the eigenvalues of degree t.With respect to the eigenbasis of the action of g on V , In fact, det(I − z • ρ(g)) is independent of the choice of basis to express ρ.Thus the generating function of characters of the symmetric powers has the rational form: For the t-th exterior power, g( The character of the t-th exterior product evaluated at g is the sum of all square-free homogeneous monomials in λ i of degree t.As such, we can rewrite We have now recovered Lemma 3.1 in [Sta11], which states: Let G be a finite group and let V be an n-dimensional representation.Then Moreover, if an element g ∈ G acts on V via a matrix A, and if I denotes the identity n × n matrix, then both sides equal 1 det(I−z•A) when the associated characters are evaluated at g.
For a G-invariant simplex S with vertices {v 0 , v 1 , . . ., v d } ∈ M × {1} as in the Setup 2.1, the infinitedimensional C-vector space C [Z ≥0 vert(S)] with basis: is isomorphic as a graded G-module to the vector space t≥0 Sym t (M ′ Z C), and their characters coincide.If S is a unimodular simplex, then (12) expresses the equivariant Ehrhart series of S. This idea is generalized to all simplices in [Sta11, Proposition 6.1], and we now generalize it further to polytopes admitting G-invariant half-open decompositions.
As in the Setup 2.1, let P be a lattice polytope with vertices in M × {1}, invariant under the linear action of a finite group G.
Proof.Let q be the number of orbits of intervals of T [ ) .There necessarily exists a G-invariant simplex S 0 contained in an interval [S 0 , S 0 ] of T [ ) that contains the G-fixed barycenter of the vertices of P .Label its vertices by {v 0,0 , . . ., v 0,d } and label the orbit of the interval [S 0 , S 0 ] by O 0 .Order the other orbits O 1 , . . ., O q−1 of intervals of T [ ) .For each i ∈ [q − 1], label representative simplices S i and S i contained in an interval [S i , S i ] of O i and label the vertices of S i as v i,0 , . . ., v i,d .Identify the Gmodule C[Z ≥0 vert(S 0 )] with the polynomial ring and graded G-module, C[X 0 , . . ., X d ] where g(X i ) := X j if g(v 0,i ) = v 0,j .Viewing g ∈ G as a permutation on {0, 1, . . ., d}, we write, g( . We define a map f from to C[Cone(P ) ∩ M ′ ] by defining f on a basis and extending linearly.The map f makes use of the expression for lattice points in Cone(P ) given in Proposition 3.24.To shorten notation, for a lattice point y 0 in Box([S 0 , S 0 ]) or gy i in Box([gS i , gS i ]), we denote the corresponding basis vector of as y 0 or gy i respectively.The direct sum ( 16) is also a graded G-module.A basis for the tensor product (15) is the set Suppose gy 0 = w 0 for box points y 0 and w 0 in Box([S 0 , S 0 ]) and therefore To check f is well-defined, we compute: The map f is an isomorphism of vector spaces (Proposition 3.24) and a G-module isomorphism, as we verify on the basis.For h ∈ G, The G-module isomorphism also respects the grading and yields an equality among characters.Identify C[X 0 , . . ., X d ] and t≥0 Sym t (M ′ Z C) as graded G modules as in the discussion around Lemma 3.26.This yields Example 3.28.[Example 3.19 continued] The two-dimensional permutahedron Π 3 ⊆ R 3 under the action of the group Z/3Z cyclically permuting the standard basis vectors admits a G-invariant half-open decomposition as dictated by Theorem 3.27.This half-open decomposition is described in Figure 6 through a triangulation T of Π 3 and a partition of the face poset of T into intervals.In this partition, each maximal simplex in an interval is a triangle and every orbit of triangles has order 1 or 3.With this half-open decomposition, the lattice points in the union of the fundamental parallelepipeds are {(0, 0, 0)} at height 0, {(1, 2, 3), (2, 3, 1), (3, 1, 2), (2, 2, 2)} at height 1, {(4, 4, 4)} at height 2.
At height 0 and 2, there is a unique Z/3Z-invariant box point, each giving a copy of the trivial representation.At height 1, the character of the permutation representation on the 4 lattice points evaluates to 4 at the identity element, and 1 at the other two group elements.Taking the inner product to express this representation in the basis of irreducible representations for Z/3Z: χ triv , χ g , χ g 2 , yields the The next theorem introduces a second way to compute the equivariant Ehrhart series using Ginvariant half-open decompositions.The conditions required by the theorem ensure that the quasipolynomial χ tP is actually polynomial in t.The equivariant Ehrhart series has the following rational generating function: d+1  .
In this case, χ tP is a polynomial in t with coefficients in R(G).
Proof.We break Cone(P ) into orbits and describe the permutation representation on each piece.Let O be an orbit of intervals of T [ ) , and let g ∈ G. Furthermore, let Cone(O) := I∈O Cone(I).
Proof of Claim.We first show that the left hand side of Equation ( 17) is a subset of the right in terms of the lattice points appearing in the exponents of the series expansions.Suppose u ∈ Cone(O) g ∩Z d+1 .By Proposition 3.22, there exists a unique interval I = [S, S] ∈ O such that u ∈ Cone(I).Let {v 1 , . . ., v n+1 } be the vertices of S. Then we may write u uniquely as u = n+1 i=1 (c i v i ) + y, with c i ∈ Z ≥0 for all i ∈ [n + 1], and y ∈ Box(I) by Proposition 3.24.As the expression of u is unique, u = g(u) = n+1 i=1 (c i g(v i ))+g(y) implies that y = g(y).This implies that the coefficient of z u in the series expansion of the right side of Equation ( 17) is 1.
We now show the right hand side of Equation ( 17) is a subset of the left.By Proposition 3.24, every lattice point u in the series expansion of the right side of Equation ( 17) as z u has coefficient one.
Figure 8: A triangulation of the permutahedron Π 3 into six triangles.
Furthermore, there exists a unique interval Thus u ∈ Cone(O) g and Equation ( 17) holds.
For all t ∈ Z ≥0 , Cone(O) g ∩ (Z d × t) is equal to the permutation character on the lattice points at height t in Cone(O) evaluated at g, which we denote by χ tO (g).Likewise, .
Summing over all the orbits yields, The only lattice points that can be box points must have a coordinate sum that is a multiple of 6.The box points for each of the respective intervals are: There are 3 orbits of box points.The character table of Z/3Z is as follows, with ζ a cube root of unity.
We calculate the following rational generating function for the equivariant Ehrhart series using the formula from Theorem 3.29: Evaluating at the group elements yields the Ehrhart series of the fixed subpolytopes.Transforming to χ tP using the usual transformation gives: Evaluating at the identity and g yields the Ehrhart polynomials of the fixed subpolytopes, 3t 2 + 3t + 1 and 1 respectively.Theorem 3.29 would need to be modified to allow for rational triangulations in order to be applied to Π 3 under the action of S 3 permuting the standard basis vectors.

Breaking Down the H*-series
In this section, we consider the hypersimplex and the (prime) permutahedra under the cyclic group action and provide an explicit formulas for the equivariant Ehrhart series.

Cyclic Group Action on Hypersimplex
We give a combinatorial interpretation (Theorem 3.33) for the H * -series of the hypersimplex ∆ k,n under the group Z/nZ where the action is given by cyclically shifting coordinates.We build upon [Kat05] to compute the H * -series and then follow the arguments in [Kim20] to give an equivariant generalization.Before stating the main theorem, we give several definitions.For two integers 0 < k < n, the (k, n)-th hypersimplex is defined to be It is an (n − 1)-dimensional polytope inside R n whose vertices are (0,1)-vectors with exactly k 1's.In particular, it has n k many vertices.A decorated ordered set partition We call each L i a block and we place them on a circle in a clockwise fashion and then think of l i as the clockwise distance between adjacent blocks L i and L i+1 where indices are considered modulo m (see Figure 9).So the circumference of the circle is l 1 + • • • + l m = k.We regard decorated ordered set partitions up to cyclic rotation of blocks (together with corresponding l i ).For example, the decorated ordered set partition ({1, 2, 7} 2 , {3, 5} 3 , {4, 6} 1 ) is the same as Given a decorated ordered set partition, we define the winding vector and the winding number.To define the winding vector, let w i be the distance of the path starting from the block containing i to the block containing (i + 1) moving clockwise (where i and (i + 1) are considered modulo n).If i and (i + 1) are in the same block then w i = 0.In Figure 9, the winding vector is w = (0, 2, 3, 3, 3, 1, 0).The total length of the path is (w 1 + • • • + w n ), which should be a multiple of k as we started from 1 and came back to 1 moving clockwise.If (w 1 + • • • + w n ) = kd, then we define the winding number to be d.In Figure 9, the winding number is 2.
We define a cyclic group Z/nZ action on the set of decorated ordered set partitions of type (k, n) by cyclically shifting elements in their parts.It is straightforward to check that the Z/nZ action cyclically shifts the winding vectors so the action preserves the winding number.
From now on we fix σ ∈ Z/nZ.We denote the order of σ with b and letting a = n b , any σ-fixed vector in R n is of the form so the number of σ-fixed lattice points of d∆ k,n equals the number of integer solutions of We denote Then ( 18) has an integer solution if and only if d is a multiple of b 1 .Suppose so, and let where the notation n k p means the coefficient of In [Kim20, Section 2.1], the second author showed that (20) can be simplified as follows: Remark 3.34.The sums in (20) and ( 21) are finite sums since n k p is zero if k is greater than n(p − 1).To simplify (21) we need the following lemma.Lemma 3.35.For positive integers a, h, i and p, we have Proof.The number a ℓp−i hp counts the number of integer solutions to equations Given a solution x 1 , . . ., x a , we write each x j as For a fixed ℓ ′ , the number of such integer solutions for (y 1 , . . ., y a , z 1 , . . ., z a ) equals Applying Lemma 3.35 to (21), we conclude Now we will count σ-fixed hypersimplicial decorated ordered set partitions of type (k, n) with winding number d.The argument follows the structure of the argument in [Kim20, Section 3], but now we must consider σ-fixed decorated ordered set partitions.Here we briefly summarize the argument.As before we fix an element σ ∈ Z/nZ and denote the order of σ with b and set a = n b .
Proposition 3.40.The number of σ-fixed hypersimplicial decorated ordered set partitions of type (k, n) with winding number d is Proof.We have A decorated ordered set partition L belongs to K d k,n (S) if and only if S is a subset of I(L).If I(L) is empty then L will be counted once when S = ∅.Now assume I(L) is non empty.Note that I(L) is σ-fixed; we denote the number of σ-orbits in I(L) with i.The number of σ-fixed S ⊆ I(L) such that |S| = j equals i j , therefore L will be counted i j times with the sign (−1) j as S ranges over all σ-fixed subsets of I(L) with S(σ) = j.We conclude that the contribution of L to (23) is m j=0 (−1) j i j = 0.So (23) equals the number of σ-fixed hypersimplicial decorated ordered set partitions of type (k, n) with winding number d.
Proof.This can be proved by the same method in the proof of [Kim20, Proposition 2.22] Corollary 3.42.Given a σ-fixed T ⊆ {1, 2, • • • , n} such that |T | = bi, we have Proof.For a σ-fixed 24), ( 25) and ( 26) Property ( 24) and (25) becomes 0 ≤ v ′ j ≤ (k − bi − 1) for all j and the property (26 Thus the solution exists only if d is a multiple of b 1 and in that case the number of such v ′ is Comparing with (22) completes the proof.

Cyclic Group Action on Prime Permutahedra
Lemma 3.44.The H * -series of the permutahedron Π n under the action of Z/nZ permuting the standard basis vectors is polynomial.
Proof.By Lemma 5.2 of [ASVM20], for σ ∈ S n with cycle type λ = (ℓ 1 , . . ., ℓ m ), H * (Π n ; z) is polynomial if and only if the number of even parts in λ is 0, m, or m − 1.For any element σ of Z/nZ, all cycles of σ have the same length, so H * (Π n ; z) is polynomial, and thus H * itself is a polynomial.
We now focus on the case when p is prime and Z/pZ acts on Π p .
Theorem 3.45.The h * -polynomial of the permutahedron Π p for prime p has coefficients equivalent to 1 mod p.
Proof.If p = 2, then h * (z) = 1.Let p > 2. Let I be the set of all nonempty linearly independent subsets of {e j − e k : 1 ≤ j < k ≤ p}, and let T be a linearly independent subset in I. Let σ = (1 2 . . .p) be a generator of Z/pZ.The group Z/pZ acts on the independent sets I by cyclically permuting the forest vertex labels.Each orbit has size p, as follows.For 1 ≤ k < p, suppose σ k T = T ∈ I. Then for each edge (i, j) in the forest corresponding to T , the edge (|i + k| p , |j + k| p ) is also in the forest.
As k is coprime to p, the forest thus contains p edges and therefore a cycle, which is a contradiction.Thus σ k does not fix any T ∈ I, and |Z/pZT | = p.Let S ⊂ I be a transversal containing one representative from each orbit in I.Each half-open parallelepiped in an orbit has the same Ehrhart series, and is translated by an integral vector.Thus, The coefficients of (1 − z) p−1 are equivalent to one mod p, because the coefficient of z i is p−1 i (−1) i , and Hence the claim follows.
Example 3.46.For the permutahedron Π 3 we have S = {{e 1 − e 2 }, {e 1 − e 2 , e 2 − e 3 }}.Thus Theorem 3.47.Let P ⊂ R n be an (n − 1)-dimensional lattice polytope invariant under the action of the cyclic group Z/nZ permuting the coordinates of R n .Furthermore, suppose that for each non-identity element g ∈ Z/nZ, P g is a single lattice point.Then, where h i is the coefficient of z i in the h * -polynomial of P , and {χ 0 , . . ., χ n−1 } are the irreducible representations of Z/nZ.
Proof.It is enough to show that our formula for H * (P ; z) specializes to the fixed Ehrhart series for each conjugacy class.Let ζ be an n-th root of unity.Then For the identity element, we have For g = id, our formula gives H * (P ; z)(g) Let Π n be the permutahedron of dimension n − 1 in R n for n prime.Let Z/nZ act on Π n by cyclically permuting coordinates of R n .
Corollary 3.48.For an odd, prime number n, the H * -series of Π n under the action of Z/nZ is where h * i is the coefficient of z i in the h * -polynomial of Π n .
We conclude our discussion of prime permutahedra by applying Stapledon's Corollary 2.10 to show the existence of a Z/pZ-invariant nondegenerate hypersurface.In Section 3.4 we continue in a similar vein, establishing a criterion that guarantees the nonexistence of such a hypersurface under the action of the symmetric group.
Theorem 3.49.Let p be prime.Then for the action of Z/pZ on Π p , there exists a Z/pZ-invariant nondegenerate hypersurface with Newton polytope Π p .Consequently H * (Π p ; z) is effective and polynomial.
Proof.If p = 2, then Π p is a line segment and the condition of Corollary 2.10 is trivially satisfied.Suppose p is an odd prime.Then every proper face of Π p with dimension > 1 corresponds to some ordered set partition of p with at least 2 parts, which do not all have the same size and hence cannot possibly be fixed under the action of any σ ∈ Z/pZ other than the identity.So the stabilizer of all proper faces is trivial, and hence all vertices are fixed under this stabilizer.Now consider the face Π p .Since n is odd, Π p contains the lattice point ( p+1 2 , . . ., p+1 2 ), which is fixed under the action of Z/pZ.Hence by Corollary 2.10, the H * -series is effective.
Remark 3.50.Theorem 3.45 combined with Corollary 3.48 gives an alternative proof of effectiveness.Since p is prime, is a nonnegative integer, so we can explicitly see H * (Π p ; z) is effective.

Describing Nondegenerate Hypersurfaces
Up until now, we have mainly focused our attention on criteria (ii) and (iii) from Conjecture 1.1.We now turn to (i), detailing what can be said about the existence of a S n -invariant nondegenerate hypersurface in the case of orbit polytopes.

Orbit polytopes
Weight polytopes arise in representation theory of semisimple Lie algebras, and the combinatorics of these polytopes have been studied a great deal in recent years [Pos09, KR12, Kha14, ACEP20, Sup20].
Here we specifically focus on weight polytopes of the Type A root system, we refer to these as orbit polytopes (also sometimes known as permutohedra).
Definition 3.51.An orbit polytope is a polytope of the form conv{σ • w : σ ∈ S n } where w ∈ Z n is any integer point, and S n acts by permuting its coordinates.
Using the machinery of Hopf monoids, it is shown in [AA17, Proposition 5.4] that the faces of generalized permutahedra are products of generalized permutahedra.It is easily shown as a corollary (see [Sup20, Example 3.5]) that the faces of orbit polytopes are products of orbit polytopes.Here, we are particularly interested in orbit polytopes which have some rectangular 2-dimensional faces.Rectangles themselves are not orbit polytopes, so the only way for a rectangle to arise as a product of orbit polytopes is as the product of exactly two line segments and any number of points.
(w 1 , w 2 , w 3 , w 4 ) (w 2 , w 1 , w 3 , w 4 ) (w 1 , w 2 , w 4 , w 3 ) (w 2 , w 1 , w 4 , w 3 ) Figure 10: A rectangular face arising as the product of two line segments.We are interested in the case where w 1 < w 2 ≤ w 3 < w 4 and w 2 − w 1 and w 4 − w 3 are odd.Then the S n -orbits of integer points in the rectangle are in correspondence with the integer points in the upper left quadrant of the figure, and each orbit consists of precisely one point in each quadrant.
Theorem 3.53.Let P be an orbit polytope which has a rectangular 2-dimensional face F with odd side lengths, and let G be the symmetric group acting by the standard representation.Then the toric variety of P does not admit a G-invariant non-degenerate hypersurface.
Proof.Let F be a rectangular 2-face of P with odd side lengths.We have established that F can be written as the product of two line segments and any number of points.When we restrict a hypersurface in the toric variety of P to the face F , the points will simply contribute a monomial factor to the expression for the hypersurface, which will not affect whether it is smooth at F .Hence we are free to ignore the points and focus our attention on the two line segments.By analyzing the linear functionals that are maximized at the face F , one can show that the two line segments must have the form conv{(w 1 , w 2 ), (w 2 , w 1 )} and conv{(w 3 , w 4 ), (w 4 , w 3 )} where w 1 < w 2 ≤ w 3 < w 4 (see Example 3.52).Since the segments must have odd length, we further know that w 2 − w 1 and w 4 − w 3 are odd.If we look at the orbits of the lattice points in F under the action of S n , we will find that each orbit contains exactly four points, with exactly one in each quadrant of F (see Figure 10).So the parameters appearing in the equation for our hypersurface restricted to F can be identified with points in one quadrant of F .We will denote them by c i,j where 0 ≤ i ≤ ⌊ w2−w1 2 ⌋ and 0 ≤ j ≤ ⌊ w4−w3 2 ⌋.Using this, a generic S n -invariant hypersurface restricted to the face F of P has the following equation: Since w 2 − w 1 is odd, then for each i one of w 1 + i and w 2 − i will be odd and the other even.Likewise, for each j one of w 3 + j and w 4 − j will be odd and the other even.Hence each term x w1+i 1 x w2−i 2 + x w2−i 1 x w1+i 2 vanishes when x 2 = −x 1 , regardless of the choices for c i,j , and each term x w3+j 3 x w4−j 4 + x w4−j 3 x w3+j 4 vanishes when x 4 = −x 3 .Because the equation has a zero of multiplicity 2 whenever x 2 = −x 1 and x 4 = −x 3 , the gradient vanishes at this locus, so no S n -invariant hypersurface in the toric variety of P is smooth at F .We will now characterize exactly which orbit polytopes have rectangular 2-dimensional faces with odd side lengths.To do this, we will associate any given S n -orbit polytope with a composition of n using the following setup.
Proof.Proposition 4.19 of [Sup20] implies that for an orbit polytope with composition α, any face can be described as a product of orbit polytopes with compositions β (1) , . . ., β (r) from which α can be obtained by some combination of concatenations and near-concatenations.Moreover, every such collection of compositions β (1) , . . ., β (r) gives a face of the orbit polytope.In order for this face to be a rectangle, we need two of the compositions β to be (1, 1) (the composition of a line segment) and the remaining β's to be (1) (the composition of a point).When such a collection of compositions is combined together with any sequence of concatenations and near-concatenations, the resulting α is as described.
Corollary 3.56.An orbit polytope with composition α arising from the point w (as in Setup 3.54), that satisfies the condition of Lemma 3.55, has a rectangular 2-face with odd side lengths if and only if one of the following criteria is satisfied: (i) There exist i and j with 1 ≤ i < i + 1 < j < j + 1 ≤ k such that the differences w i+1 − w i and w j+1 − w j are both odd.
(ii) There exists an i with 1 ≤ i ≤ k−2 such that α i+1 ≥ 2 and the differences w i+1 −w i and w i+2 −w i+1 are both odd.

Hypersimplices
We let the symmetric group S n acts on the hypersimplex ∆ k,n by permuting coordinates.The following theorem implies that H * -series of the hypersimplex ∆ k,n is effective.
Theorem 3.57.The toric variety of the hypersimplex ∆ k,n admits an S n -invariant non degenerate hypersurface.
• For p < q and i ∈ S p , j ∈ S q , c i > c j .
Let h be the integer such that h i=1 |S i | ≤ k and h+1 i=1 |S i | > k.Then vertices of ∆ k,n that maximizes L are given by (x 1 , x 2 , . . ., x n ) such that • For i ∈ S p with p ≤ h, x i = 1.
• For i ∈ S p with p > h + 1, x i = 0.
• The number of i's such that x i = 1 for i ∈ S h+1 equals (k − h i=1 |S i |).For other i's, x i = 0. We conclude that e k (x 1 , x 2 , . . ., x n )| F equals monomial times elementary symmetric functions with variables {x i } i∈S h+1 .Using the same argument, this defines a smooth hypersurface inside a torus.

Further Questions
Question 4.1.Section 3.1 uses zonotopal decompositions to compute the equivariant Ehrhart series of zonotopes of the Type A root system.How can this technique be adapted to general root systems?See [ABM20] for some progress in this direction.
Question 4.2.Can we characterize graphs Γ for which every σ ∈ Aut(Γ) satisfies one of the conditions of Theorem 3.15?Question 4.3.Theorem 1.4 of [BJM16] gives a formula for the h * -polynomial of a zonotope in terms of the basis activity of its corresponding matroid.Is there an equivariant analogue of this formula that describes the H * -series of a zonotope in terms relating to the G-action on its matroid?Question 4.4.For which n is there a G-invariant triangulation as in Theorem 3.27 or Theorem 3.29 of the permutahedron Π n ∈ R n under the cyclic group action permuting the standard basis vectors?Can Theorem 3.29 be extended to compute equivariant Ehrhart theory using symmetric zonotopal tilings?Question 4.5.Can we find large families of polytopes exhibiting the G-invariant half-open triangulations described in Theorems 3.27 and 3.29?Question 4.6.Theorem 3.33 gives a combinatorial description of the H * i characters for the action of Z/nZ on the hypersimplex ∆ k,n .However, we also know from Theorem 3.57 that the H * -series for the S n action on ∆ k,n is effective.Can we find a combinatorial interpretation of these characters H * i of S n ?Question 4.7.Theorem 3.53 shows that any orbit polytope containing a rectangular 2-face with odd side lengths does not admit an S n -invariant non-degenerate hypersurface.Is the converse true as well?In other words, are rectangular 2-faces with odd length sides the only obstruction to the existence of an S n -invariant nondegenerate hypersurface in the toric variety of an orbit polytope?Question 4.8.Suppose that the G-invariant polytope P admits an invariant non-degenerate hypersurface, or has polynomial/effective H * -series.Must the same then be true for the action of G on positive integer dilations of P ?Table 1 summarizes what we know about the action of S 4 on the dilates of Π 4 .Question 4.9.Although condition (i) of Conjecture 1.1 was proven to not be implied by conditions (ii) and (iii) (Theorem 1.2), it may still be the case that (ii) and (iii) are equivalent.Does polynomiality of the H * -series imply effectiveness in general?Furthermore, what additional hypotheses need to be added to Conjecture 1.1 in order to make it true?Question 4.10.A topic of great interest in classical Ehrhart theory is determining which polytopes have h * -vectors that are unimodal ; i.e., each coefficient is weakly larger the previous, until a peak is reached, after which point each coefficient is weakly smaller than the previous.In the equivariant setting, we can interpret unimodality by looking at the differences H * i − H * i-1 and asking that these differences form a sequence consisting of effective characters up to a certain point, after which all the differences are negatives of effective characters.Which polytopes exhibit this equivariant unimodality property, and what parallels can be drawn with the classical concept of h * -unimodality?Question 4.11.The Ehrhart f * -vector is a change-of-basis transformation of the h * -vector, and f *positivity for polyhedral complexes is a weaker property than h * -positivity.What is the equivariant analogue of the f * -vector, and what can we say about its effectiveness?This question was suggested by Katharina Jochemko.

Acknowledgements
We are grateful to Paco Santos and Alan Stapledon for sharing with us the counterexample of Theorem 1.2, and for a great deal of helpful email correspondence.We also thank Vic Reiner for proposing this problem and Federico Ardila, Matt Beck, Christian Haase, Katharina Jochemko, and Kris Shaw for fruitful conversations.Jean-Philippe Labbé provided useful advice in the development of the Sagemath package of Appendix A. Katharina Jochemko, as well as the Discrete Geometry group at FU Berlin, contributed travel funding for research visits that were instrumental in this project.

Figure 1 :
Figure1: Action of (Z/2Z) 2 on the 3-cube.Each of σ, τ , and στ acts by a rotation of 180 degrees around a corresponding axis.The vertices are divided into two orbits under this action, which are indicated by their shapes.

Figure 2 :
Figure 2: (2a) The first three dilates of P and (2b) the cone over P .

Figure 5 :
Figure 5: The connectivity graphs C Path n (σ) where σ is the generator of Z/2Z in the case where n is even (5a) and odd (5b).
m i=1 (1 − z ℓi ) has a zero at z = −1 of order equal to the number of even ℓ i , condition (c) is exactly what is needed for cancellation to occur.Corollary 3.16.H * (Z Γ ; z) is a polynomial if and only if every σ ∈ Aut(Γ) satisfies one of the conditions of Theorem 3.15.

Figure 6 :Figure 7 :
Figure 6: A half-open decomposition of the permutahedron Π 3 determined by a triangulation (top image) and a partition of the face poset of the triangulation into intervals shown via its poset diagram (bottom image)

Definition 3. 23 .
Let P ⊆ R d be a lattice polytope, and let I = [S, S] be an interval in a half-open decomposition T [ ) of P .The half-open fundamental parallelepiped of the interval I, denoted (I), is the set of points x ∈ R d+1 such that:x = vi∈S λ i (v i , 1) + vj ∈S\S µ j (v j , 1), where λ i ∈ (0, 1] and µ j ∈ [0, 1).For an interval I of a half-open decomposition, Box(I) is defined to be the set of lattice points in the half-open fundamental parallelepiped (I): Box(I) = (I) ∩ Z d+1 .Throughout, we use Box(I) k to denote the set of lattice points in the half-open fundamental parallelepiped of I at height k, i.e. with last coordinate equal to k. Proposition 3.24.Let P be a lattice polytope with half-open decomposition T [ ) .Every lattice point x ∈ Cone(P ) can be expressed uniquely as x = w + y for some w = vi∈S λ i (v i , 1), with λ i ∈ Z ≥0 and y ∈ Box(I) for some interval I = [S, S] ∈ T [ ) .Proof.Let x ∈ Cone(P ).Then by Proposition 3.22, x ∈ Cone(I) for a unique interval I = [S, S] of T [ ) , and Definition 3.25.Let P be a lattice polytope invariant under the action of a group G.A half-open decomposition T [ ) of P is called G-invariant if 1. Simplices are sent to simplices (the triangulation is G-invariant), and 2. The action of G induces an automorphism of the face poset of T [ ) such that intervals of T [ ) are sent to intervals.
For a half-open decomposition T [ ) of P , let Box(T [ ) ) i denote the union of the sets Box(I) i for intervals I of T [ ) .The group G permutes the lattice points in Box(T [ ) ) i , and the corresponding permutation character is written χ Box(T [ ) )i .Theorem 3.27.Let T [ ) be a G-invariant half-open decomposition of a d-dimensional polytope P as in the Setup 2.1 such that dim(S) = d for all intervals [S, S] of T [ ) and such that all orbits of intervals of T [ ) have order |G| except for a unique G-invariant interval.Then the H * -polynomial is the permutation representation on the union of box points in T [ ) , graded by height: For a G-invariant half-open decomposition T [ ) of a polytope, let O denote an orbit of intervals of T [ ) , dim(O) := dim(S) for any interval I = [S, S] ∈ O, and Box(O) i := I∈O Box(I) i .Let χ Box(O)i denote the permutation character on the lattice points in Box(O) i .Theorem 3.29.Let P be a G-invariant d-polytope as in the Setup 2.1 with a G-invariant half-open decomposition T [ ) such that no interval consists of a single d-face of the triangulation.Furthermore, suppose that for each interval I = [S, S] of T [ ) and each g ∈ G, if a box point y ∈ Box(I) is fixed by g, then the simplex S is fixed pointwise by g.

Example 3. 31 (
Example 3.28 continued).Again consider the two dimensional permutahedron Π 3 ⊆ R 3 under the action of Z/3Z cyclically permuting the standard basis vectors.It is necessary to use a different half-open decomposition from that in Example 3.28 to compute the equivariant Ehrhart series.As shown in Figure 8, we triangulate Π 3 using a barycentric subdivision.The intervals we use for a G-invariant half-open decomposition are: [∅, g], [a, abg], [b, beg], [c, acg], [d, cdg], [e, ef g], [f, df g].

Figure 9 :
Figure 9: The figure on the left is the picture associated to the decorated ordered set partition ({1, 2, 7} 2 , {3, 5} 3 , {4, 6} 1 ) and the figure on the right is the picture after acting by σ.Now we state our main theorem in this section.Theorem 3.33.The character H * d (∆ k,n ; z) is the permutation character on the set of decorated ordered set partitions of type (k, n) with winding number d.

s a g
e : Pi4 = p o l y t o p e s .permutahedron ( 4 , backend =' n o r m a l i z ' ) s a g e : Pi4 A 3− d i m e n s i o n a l p o l y h e d r o n i n ZZˆ4 d e f i n e d a s t h e conv ex h u l l o f 24 v e r t i c e s s a g e : G = Pi4 .r e s t r i c t e d a u t o m o r p h i s m g r o u p ( output =' permutation ' ) s a g e : G Permutation Group with g e n e r a t o r s [ ( 1 , 6 ) ( 3 , 1 2 ) ( 4 , 8 ) where Φ, Ψ H ∈ C denotes the inner product between characters Φ, Ψ of H, and χ i | H is the representation restricted to H.Example 2.14.The cyclic group Z/nZ ⊆ S n acting on the permutahedron Π n gives an interesting example of the failure of the other direction.Namely, H * Sn (Π n ; z) is non-polynomial for n ≥ 4 [ASVM20], but H * Z/nZ (Π n ; z) is polynomial for all n (see Lemma 3.44).Here we show how the rational function collapses to a polynomial for the case H = Z/4Z ⊆ S 4 : We implemented functionality in Sagemath for the calculation of the H * -series, and we use it to calculate H * (Π 4 , ; z) below, with characters given in the following tables.See Appendix A for details.

.
Proof of Theorem 3.33.By Proposition 3.40 and Corollary 3.42, the number of σ-fixed hypersimplicial decorated ordered set partitions (of type (k, n) with winding number d) is

Table 1 :
Conditions of Conjecture 1.1 when S 4 acts on dilates of Π 4 .