On the Todd Class of the Permutohedral variety

In the special of braid fans, we give a combinatorial formula for the Berline-Vergne's construction for an Euler-Maclaurin type formula that computes number of lattice points in polytopes. By showing that this formula does not always have positive values, we disprove a positivity conjecture we made previously. Our combinatorial formula is obtained by computing a symmetric expression for the Todd class of the permutohedral variety. Also as a result , we prove that the Todd class of the permutohedral variety $X_d$ is not effective for $d \ge 24$. Additionally, we prove that the linear coefficient in the Ehrhart polynomial of any lattice generalized permutohedron is positive.


Introduction
Let V be a real finite dimensional vector space and Λ ⊂ V a lattice. A lattice polyope in V is a polytope such that all of its vertices lie in Λ. A classical problem in the crossroads between enumerative combinatorics and discrete geometry is that of counting lattice points in lattice polytopes. For any polytope P ⊂ V we define Lat(P ) := |P ∩ Λ|. One of the earliest results in the area is Pick's theorem, which says that for any Z 2 -polygon P ⊂ R 2 we have Lat(P ) = a(P ) + 1 2 b(P ) + 1, where a(P ) is the area of P and b(P ) is the number of lattice points in the boundary of P . One way to obtain a higher dimensional analogous of Pick's formula is to find a formula relating the amount of lattice points of P with the different normalized volumes of the faces F of P . We want a real-valued function α on pairs (F, P ), where F is a face of a lattice polytope P , such that where nvol(F ) is the normalized volume of F . It is clear that for a given lattice polytope P one can always find many functions satisfying (1.1). What we want is a function that works simultaneously for all lattice polyopes. We can do this by requiring the function α to be local, i.e., if the numbers α(F, P ) only depend on the local geometry of P around F , or more specifically, the value only depends on ncone(F, P ), the normal cone of P at F. Any local function α that satisfies Equation (1.1) for all lattice polytopes P is called a McMullen function, since McMullen was the first to prove their existence [13]. His proof is nonconstructive and shows that there are infinitely many McMullen functions. In the present paper we compute the values for a particular McMullen function on a special family of polytopes: generalized permutohedra, defined and thoroughly studied by Postnikov in [15]. Our methods are based on the theory of toric varieties.
1.1. Todd classes of toric varieties. Let P be a lattice polytope with normal fan Σ and X Σ be the associated toric variety. The Todd class Td(X Σ ) is an element in the Chow ring of X Σ . As such it can be written as a Q-linear combination of the toric invariant cycles [V (σ)]: Since the cycles [V (σ)] satisfy algebraic relations, the values r Σ (σ) satisfying (1.2) are not uniquely determined. An amazing connection with lattice polytopes is given by the fact that any function r Σ (·) satisfying (1.2) defines a function α satisfying (1.1) for P by setting α(F, P ) = r Σ (ncone(F, P )). A proof of this fact can be found in Danilov's 1978 survey [5] where he further asked if there exist a function r that depends only on the cone σ and not on Σ, in other words, if there exist a local function r satisfying Equation (1.2) for all fans Σ. Accordingly, we call such a function r on pointed cones a Danilov function. By setting α(F, P ) = r(ncone(F, P )) any Danilov function gives a McMullen function. We want to briefly remark on two constructions of Danilov functions from the last fifteen years. Pommersheim and Thomas [14] gave a construction of a Danilov function r(σ) that depends on choosing a complement map for subspaces. Originally they do this by choosing a complete flag of subspaces. One technical issue is that their construction of r(·) is not defined for all cones σ, it requires σ to be generic with respect to the chosen flag. A couple of years later Berline and Vergne [2] constructed a McMullen function with the property that it is computable in polynomial time fixing the dimension and it is a valuation on cones. We call this construction the BV-function, and denote it by α bv . Later in [1], they showed that their construction is actually a Danilov function. Therefore, we abuse the notation, and consider α bv is both a function on pairs (F, P ) and a function on cones with the connection that α bv (F, P ) = α bv (ncone(F, P )).
In [10] Pommersheim and Garoufalidis proved that using an inner product for a complement map in the methods of [14] results in the Danilov function α bv , which in turns gives an alternative way of computing it.
Both constructions, Berline-Vergne's and Pommersheim-Thomas', are algorithmic. A priori it is very hard to get formulas for general cones. There are very few examples of fans Σ for which α bv (σ) (or any other Danilov function) have been computed for all σ ∈ Σ. The main contribution in the present paper is a combinatorial formula for α bv on all cones in braid fans. (See Section 2.1 for the definition of braid fans.) In [3] we exploited an extra symmetry property satisfied by the function α bv , and use this symmetry to study the values on cones in braid fans. The main theorem in [3] is the uniqueness theorem, which in the context of the present paper states that, for the specific example of braid fans, α bv is the unique function satisfying Equation (1.2) and being invariant under the permutation action of the symmetric group on the ambient space. This is the main tool that will allow us to compute the BV-function.
1.2. Connection to Ehrhart theory. In [7] Ehrhart proved that for every lattice polytope P the function Lat(tP ), t ∈ N is a polynomial in t of dimension d = dim P , i.e., Lat(tP ) = a 0 + a 1 t 1 + a 2 t 2 + · · · + a d t d , a i ∈ Q.
The right hand side is called the Ehrhart polynomial of P . Given a McMullen formula α one can deduce that We call a lattice polytope P Ehrhart positive if all the coefficients of its Ehrhart polynomial are positive (see [12] for a recent survey on Ehrhart positivity). A consequence of Equation (1.3) is that if we have a McMullen function α such that α(F, P ) is positive for all faces F ⊂ P then P is Ehrhart positive. The converse is not true as shown in Section 3.4 of [4]. In [6] the authors conjectured that matroid polytopes are Ehrhart positive. One of the main motivations for [3] was trying to prove that conjecture. Matroid polytopes are generalized permutohedra and we focus on that larger family of polytopes. As an attempt to prove Conjecture 1.1 we proposed also the following stronger conjecture. Conjecture 1.2 (Conjecture 1.3 [3]). Let P be a generalized permutohedron and F ⊂ P a face, then α bv (F, P ) is positive. In other words, since α bv (F, P ) = α bv (ncone(F, P )), the conjecture is equivalent to saying that α bv is positive on every cone in the braid fan.
In the present paper we use our formula in Theorem 4.4 to find a negative value for α bv on cones in braid fans, hence disproving Conjecture 1.2. This does not imply that Conjecture 1.1 is false, and in fact we present a proof, independent of the rest of the paper, that the linear coefficient of the Ehrhart polynomial of any lattice generalized permutohedron is positive. This result was proved independently by Jochemko and Ravichandran in [11], using different techniques from what are presented in this paper.
Finally, as a consequence of these negative values we also obtained the following result about the permutohedral variety. That is, there is no way of expressing it as a nonnegative combination of cycles.

1.3.
Organization. This paper is organized as follows. In Section 2 we set the preliminaries. Our definition of the Chow ring is quite elementary, for a more in depth treatment of the subject of toric varieties see [9]. In Section 3 we define combinatorial diagrams that will be used to express formulas asserted in Theorem 4.4. In section 4 we present our explicit general formula followed by some applications in Subsection 4.1. Finally in Section 5 we prove that the linear coefficient in the Ehrhart polynomial of any lattice generalized polyhedron is positive, providing an evidence to Conjecture 1.1.

Acknowledgements
The second author is partially supported by a grant from the Simons Foundation #426756. This project started when both authors were attending the program "Geometric and Topological Combinatorics" at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester, and they were partially supported by the NSF grant DMS-1440140.
We wish to thank Jamie Pommersheim for many helpful discussions.

Preliminaries and notation.
We assume familiriaty with the concepts of polytopes, normal fans, and toric varieties. A good reference is [8]. Here we review concepts and notation that we are going to use. As standard we denote [d + 1] := {1, 2, 3, · · · , d, d + 1}. The set of all subsets of [d + 1] form a poset B d+1 called the boolean algebra and we define the truncated boolean algebra, denoted by B d+1 , to be the poset obtained from B d+1 by removing [d+1] and ∅. Two elements A k-chain S • = (S 1 , · · · , S k ) is a sequence of k totally ordered elements of B d+1 . For notational purposes, we complete S • by adding ∅ and [d + 1] to obtain In particular, if v = (1, 2, . . . , d + 1), we obtain the regular permutohedron, denoted by Π d , Π d := Perm(1, 2, . . . , d + 1). As long as there are two different entries in v we have dim(Perm(v)) = d. A generic permutohedron is any polytope of the form Perm(v) where all the entries of v are distinct.
Here we consider instead the translation Π d : Now we define one of our central objects, the braid fan Σ d . This fan can be combinatorially described as follows. Let e 1 , · · · , e d+1 be the standard basis of R d+1 and for each S ∈ B d+1 define e S := i∈S e i as an element in W d . For any k-chain S • of B d+1 , we define the corresponding braid cone σ S• := Cone(e S : S ∈ S • ), which is k-dimensional. The following is well known. Any polytope whose normal fan is a coarsening of Σ d is called a generalized permutohedra. These polytopes were defined and studied initially in [15]. Matroid polytopes are the generalized permutohedra for which every vertex has coordinates in {0, 1}.

Permutohedral variety.
The permutohedral variety X d is the toric variety associated to Σ d . Each cone σ S• , with § • ∈ C d+1 , of the braid fan has a subvariety V (σ S• ) associated. These subvarieties are the torus invariant cycles.
For any d ∈ N we define the following ring R d := k[x S : S ∈ B d+1 ], where k is any algebraically closed field. For any element i ∈ [d + 1] we define the linear form ℓ i := S∋i x S . Definition 2.3. The Chow ring of the permutohedral variety X d can be presented as where We are interested in computing the Todd class of X d in A d . Following [9, Section 5] the Todd class can be defined as follows.
which is an element of A d by expanding each parenthesis on the right hand side as Here B i is the i-th Bernoulli number and note that f k = 0 for any k > d and f ∈ A d so the sum in (2.3) is finite.
The class of the subvariety and it can be represented in A d as where S • = (S 1 , · · · , S k ). We are interested in expressions for Td(X d ) in terms of the torus invariant classes. In other words, we are looking for r(S • ) ∈ Q such that Remark 2.5. By Equation (2.4) an expression of the form (2.5) can be obtained by finding a square-free representation in A d .
Our interest in such expression lies in the following theorem originally atributted to Danilov already mentioned in the introduction. Here we only state it in the particular case of braid fans.
Theorem 2.6 (Section 5 in [9]). Let P be a d-dimensional lattice generalized permutohedron with normal fan Σ d . For any expression as in Equation (2.5), we have that In other words, an equation of the form (2.5) gives a solution to (2.6) for lattice generalized permutohedra by setting α(F, P ) = r d (ncone(F, P )).
We are focusing on the particular case of braid fans and not on all possible fans at the same time, so a priori we are not looking for a Danilov function. However, we are going to require one more special for our expressions of the form (2.5).
Definition 2.7. The symmetric group S d+1 acts on elements of B d+1 hence on the generators of the ring R d . Notice that this action fixes both ideals I 1 and I 2 so that S d+1 acts naturally on A d too. We say an element f ∈ A d is symmetric if π · f = f for all π ∈ S d+1 . Any element f can be symmetrized: We want a square-free symmetric expression for Td(X d ) in the presentation (2.1) of A d .
Remark 2.8. In the ring A d , any square-free element is of the form S•∈C d+1 r(S • )x S• and it is symmetric if and only if the r(S 1 , · · · , S k ) depends only on the sequence of integers Using results from [3] we prove the following. (Abusing notation and we let α bv (S • ) = α bv (σ S• ).) Theorem 2.9. [Theorem 5.5 in [3]] There is a unique symmetric expression for Equation (2.5). This is given by the Berline-Vergne construction: Proof. By Theorem 2.6, any expression for Equation (2.5) yields a an expression Equation Combining the theorem with the symmetrization described in Equation (2.7) we get the following Proposition 2.10. Let f be any square-free expression for Td(X d ) (as in Equation (2.5)), then f ♯ is the α bv expression, i.e., the left hand side of Equation (2.8).

Spider diagrams
In this section we develop the necessary combinatorial language to express our main formulas in Section 4. Definition 3.1. Let A be a totally ordered set. A spider Sp(A, h) is a graph with vertex set A with a marked vertex h ∈ A, called the head, and an edge between every other vertex and the head. Any vertex which is not the head is called a leg. Since A is totally ordered, we can partition the set of the legs into two subsets, Left and Right, according to whether they are less than or greater than the head. A Left leg will be labeled as a L i if it is the i-th smallest vertex among all Left legs, and a Right leg will be labeled by a R j if it is the j-th largest vertex among all Right legs. If there are no Left legs, we give the head vertex h an additional label a L 1 ; similarly, if there are not Right legs, we give the head vertex h an additional lable a R 1 . See Figure 1 for an example of a spider. A graph on a single vertex can be regarded as a trivial spider by declaring the vertex its head and having no legs. Let A be an ordered set. A spider diagram D(A) on A consist of a partition of A into intervals A 1 , · · · , A k with |A i | = m i , together with a spider structure Sp(A i , h i ) on each A i and additionally we always add two extremal trivial spiders Sp 0 and Sp k+1 at the beginning and end. The head set Heads(D) is the set {h 1 , · · · , h k }, in the head sets we ignore the extremal heads. The spiders are naturally labeled Sp i for i = 1, . . . , k. The legs are now triply indexed: the element a P i,j with P ∈ {L, R} is the jth smallest/largest on the side L/R of the i-th spider. See Figure 2 for an example of this indexing system. In all of our applications the ordered set A will consist of chains of subsets. Let S • ∈ C k d+1 be a k-chain of subsets. The set of all spider diagrams having S • as heads is denoted SpD(S • ).
Definition 3.2. Let T • ∈ C n d+1 be an n-chain and let D be a spider on T • with spiders Sp 0 , Sp 1 , · · · , Sp k , Sp k+1 and head set S • for some S • ∈ C k d+1 . We define the internal weight of a single spider Sp i as and the boundary weight of the diagram D as Notice that the internal weights of the extremal spiders is 1.  3. An example of D ∈ LSpD((S 1 , S 2 ), 12) is given in Figure 3.3 together with the two extremal trivial spiders. Its weight is .
Notice how the weight of Sp 3 is just its boundary term which is equal to 1.

General formula
We are looking for square-free expressions in A d . We start by dealing with squares.
Proof. Using the relation ℓ a − ℓ b ∈ I 2 we get x S (ℓ a − ℓ b ) = 0 in A d . Expanding we get The relations in I 1 imply that any term in the parenthesis with neither T ⊆ S nor S ⊆ T has zero product with x S , thus we expand the above equation to get (4.3) T ⊆S,a∈T The third term is zero since the condition is vacuous. In the second term notice that the condition a ∈ T is redundant. After canceling terms from the second and fourth sums, everything reduces to By solving for the only appearance of x 2 S we get Equation (4.1).
The square-free expression obtained in Lemma 4.1 is not symmetric. To adjust this, we average over all possibilities.
Lemma 4.2. Let S 1 , · · · , S k be a k-chain in B d+1 and S l a set in the chain. We have the following equality in A d : We expand x 2 S l as in Lemma 4.1 using all possible pairs (a, b) ∈ (S l −S l−1 )×(S l −S l−1 ) and take the average. For the first sum, notice that the set T will appear whenever a ∈ T −S l , since each a must be paired with some b, each monomial appears |T − S l−1 | · |S l+1 − S l | times. When averaging we divide by |S l − S l−1 | · |S l+1 − S l |, thus we get the desired coefficient. The second sum is analogous.
Repeated use of this lemma allow us to expand any monomial as a sum of squarefree monomials. Proposition 4.3. Let S • ∈ C k d+1 be a k-chain, then we have the following equality in the ring A d : Proof. We follow by induction the base case being m i = 1 for all i. In this case we have the square-free monomial k i=1 x S i and LSPD(S • , k) consists of a single diagram with k trivial spiders (plus the two extremal ones). In this case sgn(D) = 1 and wt(D) = 1 so Equation (4.5) is trivially true.
For the inductive step we multiply each extra variable one by one. Let j = max{i : m i > 1}. By induction hypothesis we can assume that we have a square-free expression We need to compute x S j · x D , knowing that x D includes x S j . We use Lemma 4.2 to expand each of this products. This process creates an extra leg in the spider Sp j for any such D ∈ LSPD(S • , m − 1). We mark the edge of this new leg by the largest possible label. Each application of Lemma 4.2 changes the sign by an extra (−1) factor, also the Lemma 4.2 makes some new coefficients appear but those are precisely accounted for the weight of the spider diagram, this is all follows from the definitions whenever this new leg is neither the first left/right leg. In the case when we add the first left/right leg, the boundary weight changes so more care is needed.
Suppose that the spider Sp j has no left legs. Then its contribution to the boundary weight is Now we introduce the first left leg: T L j,1 . Lemma 4.2 introduces a coefficient of the formula continue to be correct when we introduce a left leg for the first time. Now let's suppose Sp j has no right legs. By the definition of j, Sp j+1 has no legs at all so the boundary term of Sp j+1 is equal to 1. Now we introduce the first right leg: T R j,1 . Lemma 4.2 introduces a coefficient of Since by convention S j+1 = T L j+1,1 , then the formula for the boundary continue to be correct when we introduce a right leg for the first time.
With Proposition 4.3 in hand, we can now write down square-free expressions of any element in A d , in particular of the Todd class: we use Equation (4.5) and collect the terms.  Proof. We first expand the Todd class using Equation (2.3). Each monomial k i=1 x m i S i appears with some coefficient in this expansion, this coefficient is Tdcoeff(D). Then we express each of these monomials using Proposition 4.3. Note how the labels of the edge was necessary in the proof for keeping track of the order in which we introduce the legs, but in general we can ignore the labels and consider all possible orderings, that is the term Binom(D). The other two terms are what we obtained by collecting all terms where x D appears.
To finish note that the definition of weight depends directly on the sizes of the element in the chain. So by Remark 2.8 each use of Proposition 4.3 produces a symmetric square free expression. By Theorem 2.9 there is only one such expression, hence (4.7).
The conclusion follows since the coefficients of a rational function are asymptotically the powers of the largest root of the denominator. Proposition 4.6 (Codimension 2 cones.). Let (S 1 , S 2 ) ∈ C d+1 be an arbitrary 2-chain, then where |S i | = s i for i = 1, 2.
Proof. We use Theorem 4.4. In this case all possible spider diagrams are shown in  Formula (4.8) (and a similar one for three dimesional cones) was already obtained in [3] relying on some general formulas in the Berline-Vergne constructions. Since there is no simple closed formula for their construction for unimodular cones of dimension larger than three, we couldn't push it further than that. The next proposition shows Theorem 4.4 in action. This formula couldn't be obtained with the previously known tools.
Proof. We use Theorem 4.4 once again. In this case all possible spider diagrams are shown in Figure 4.1.  Example 4.8 disproves Conjecture 1.2. Furthermore, it also enable us to prove Theorem 1.3, which we restate here.
Theorem 4.9. The Todd class of the permutohedral variety X d is not effective for d ≥ 24. That is, there is no way of expressing it as a nonnegative combination of the cycles.
Proof. First note that in the Chow ring of a toric variety arbitrary cycles can be expressed as positive combinations of torus invariant cycles, so it suffices to show that there is no positive expansion using torus invariant cycles, i.e., that there is no expression of the form (1.2) with all coefficients positive.
By Proposition 2.10, if there is any positive square-free expression for the Todd class of X d , then α bv (·) is positive for all C d+1 , but Example 4.8 shows that this is false for d = 24. Moreover, Remark 3.6 in [3] implies that there are negative values for all d ≥ 24.

Edge positivity
As mentioned in the introduction for every lattice polytope P the function Lat(tP ), t ∈ N is a polynomial in t of dimension d = dim P , i.e., Lat(tP ) = a 0 +a 1 t 1 +a 2 t 2 +· · ·+a d t d , a i ∈ Q. This is the Erhhart polynomial of P and will be denoted Lat(P, t). We also define Lat i (P ) := [t i ]Lat(P, t), the coefficient of t i in the Ehrhart polynomial. 5.1. α bv positivity. In this section we take a different argument to show that α bv values are indeed positive on codimension one cones in the braid fan and thus the main conjecture 1.1 is true for Lat 1 . The arguments in this section are independent of the rest of the paper. We make use of special polytopes called hypersimplices.
Proof. This is a consequence of [3,Theorem 5.5]. In the case of an edge the mixed valuation is equal to the valuation itself, the rest of the formula is positive hence the first part follows. The second part is a consequence of the reduction theorem [3, Theorem 3.5] which shows how the positivity of α bv for all codimension k cones in Σ d implies positivity of Lat k for all generalized permutohedra.
Proposition 5.3. The Ehrhart polynomial for ∆ k,d+1 is given by This formula can be turned into the more explicit