Lattice polytopes from Schur and symmetric Grothendieck polynomials

Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the $h^\ast$-vector in the case of Schur polynomials.


Introduction
Of central interest in algebraic combinatorics are polynomials f ∈ C[x 1 , x 2 , . . . , x m ], which commonly appear as generating functions that encode some combinatorial information. Associated to each polynomial f is the Newton polytope Newt(f ), which is the convex hull of the exponent vectors occurring in the monomials in f . A polynomial has saturated Newton polytope if every lattice point appearing in the Newton polytope corresponds to the exponent vector of a monomial in f with nonzero coefficient [19].
If a polynomial f has saturated Newton polytope, then checking if a monomial has nonzero coefficient is equivalent to checking if the corresponding integer lattice point is in the Newton polytope. Adve, Robichaux and Yong [1] use this perspective to study the computational complexity of the "nonvanishing problem" for polynomials, with a focus on Schubert polynomials.
In this paper, we study Newton polytopes arising from Schur polynomials and a generalization of symmetric Grothendieck polynomials, which we call inflated symmetric Grothendieck polynomials. We denote these polytopes by Newt(s λ ) and Newt(G h,λ ), respectively. The polytopes Newt(s λ ) and Newt(G 1,λ ) have previously been studied by Monical, Tokcan, and Yong [19] and Escobar and Yong [12], but many open questions remain. We are particularly interested in determining when Newt(s λ ) and Newt(G h,λ ) are reflexive and when they have the integer decomposition property (IDP), both of which we define in Section 2.
The Ehrhart series of a lattice polytope P is a combinatorial tool that enumerates the lattice points in dilations of P. The h * -vector of P, denoted h * (P), records the coefficients

Convex polytopes and Ehrhart theory
A polytope P ⊂ R m is the convex hull of finitely many points v 1 , . . . , v k ∈ R m . That is, The inclusion-minimal set V ⊆ R m such that P = conv(V ) is called the vertex set of P. A polytope is called lattice (resp. rational) if P = conv(V ) for V ⊆ Z m (resp. V ⊆ Q m ). Given a polytope P ⊆ R m , the classical Minkowski-Weyl theorem states that we can express P as a bounded set of the form where a i , x = m j=1 a ij x j , for some a 1 , . . . , a ∈ R m and b 1 , . . . , b ∈ R. If none of these constraints are redundant, each constraint defines, by equality, a facet (i.e., codimension 1 face) of P. The dimension of P, denoted dim(P), is defined to be the dimension of its affine span in R m .
Let P be a lattice polytope with dim(P) = d m. Given a positive integer t, let tP := {tx | x ∈ P} be the t-th dilate of P. The lattice point enumeration function is called the Ehrhart polynomial of P. By a classical result of Ehrhart [11], this function agrees with a polynomial of degree d in the variable t. Equivalently, one may also consider the Ehrhart series of P which is defined to be the formal power series Ehr P (z) := 1 + The numerator of the Ehrhart series is called the h * -polynomial and the vector of coefficients h * (P) = (1, h * 1 , . . . , h * d ) the h * -vector. By a result of Stanley [27], (1, h * 1 , . . . , h * d ) ∈ Z d+1 0 . Studying h * (P) often informs the algebraic and geometric structure of a lattice polytope P.
If 0 is in the interior of P ⊂ R m , the (polar) dual polytope of P is the polytope P * := {y ∈ R m | y, x 1 for all x ∈ P} .
A polytope P with 0 in its interior is called reflexive if P * is a lattice polytope.
Hibi ([15]) showed that P ⊂ R m with 0 in its interior is reflexive if and only if for any facet P ∩ {x ∈ R m | a, x = b} where a is primitive (meaning the greatest common divisor of the coordinates of a is 1) and b > 0, we have b = 1. In this case, there are no lattice points between the hyperplane spanned by the facet and its translation through 0, and we say that 0 is lattice distance 1 from the facet.
We will use the above characterization to extend the notion of reflexivity to polytopes that are not full-dimensional or for which the polytope has a nonzero lattice point in the relative interior. We say that a k-dimensional polytope P with a point p in its relative interior is reflexive if there is a lattice-preserving linear transformation and translation by −p that takes P to a reflexive polytope in R k . This can be tested by checking that the lattice point p is lattice distance 1 from all the facets of P, that is, that there are no lattice points in aff(P) between the span of the facet and its translation containing p.
Theorem 2 (Hibi [15]). Let P be a lattice polytope of dimension d containing the origin in its interior and having Ehrhart series Then P is reflexive if and only if h * In other words, the h * -polynomial of a reflexive lattice polytope is a palindromic polynomial of degree d.
A relaxation of reflexivity is the Gorenstein property. We say that P is Gorenstein if there is some positive integer c such that cP is a reflexive polytope, and the integer c is called the Gorenstein index of P. Similarly, this is completely detected by the Ehrhart series, as P is Gorenstein if and only if its h * -polynomial is palindromic of degree d − c + 1 by a result of De Negri and Hibi [10].
Given a lattice polytope P, one can consider the interplay between the convex geometry of P in R m with the induced arithmetic structure of P ∩ Z m . This motivates the discussion of triangulations and the integer decomposition property. A (lattice) triangulation T of P is a decomposition of P as a lattice simplicial complex. We say that T is regular if the triangulation is induced as the domains of linearity of a piecewise-linear, convex function σ : P → R. We say that T is unimodular if each maximal simplex ∆ ∈ T is a unimodular simplex, that is, if the vertices of ∆ generate Z d . We say that P has the integer decomposition property (IDP) if for any positive integer t and any lattice point p ∈ tP ∩ Z m , there are t lattice points v 1 , . . . , v t ∈ P ∩ Z m such that p = v 1 + · · · + v t . The existence of a (regular) unimodular triangulation of P ensures that P has IDP. This implication is strict, as one can construct examples of polytopes with IDP without a unimodular triangulation (see, e.g., [8,13]).
A sequence a 0 , a 1 , . . . , a n of real numbers is unimodal if there is some 0 j n such that a 0 a 1 · · · a j−1 a j a j+1 · · · a n . A common investigatory theme in Ehrhart theory is determining under what conditions one may ensure that coefficients of the h * -vector form a unimodal sequence. The most notable sufficient result is the following.
Given that these conditions are rather restrictive, it is natural to consider relaxations to determine if unimodality still holds. It is known that Gorenstein is not sufficient for unimodality as indicated by Payne [21], though none of the examples from this reference exhibit IDP.
The following even broader question was posed by Scheppers and Van Langenhoven: Question 4 (Scheppers and Van Langenhoven [23]). If P has the integer decomposition property, is h * (P) a unimodal sequence?

Newton polytopes
A polynomial f has saturated Newton polytope (SNP) if every lattice point α ∈ Newt(f ) ∩ Z m appears as an exponent vector of f , that is, c α = 0. This notion was introduced by Monical, Tokcan, and Yong in [19].
We now define our main objects of study in this paper, Newton polytopes arising from Schur polynomials and from inflated symmetric Grothendieck polynomials, both of which have SNP. A partition of a nonnegative integer n with at most m parts is λ = (λ 1 , λ 2 , . . . , λ m ) with λ 1 λ 2 · · · λ m 0 and m i=1 λ i = n. This is denoted by λ n. The number of positive parts of λ is denoted by (λ). The Young diagram associated to λ is an arrangement of boxes with λ i boxes in the i-th row, with rows aligned at the left. Given partitions µ and λ such that the Young diagram of λ is contained in the Young diagram of µ, the skew shape µ/λ is the Young diagram consisting of boxes in µ which are not in λ. A semistandard Young tableau is a filling of a Young diagram with positive integers such that entries are weakly increasing along each row and strictly increasing along each column. Let SSYT [m] (µ/λ) denote the set of all semistandard Young tableaux of shape µ/λ with fillings from [m] = {1, . . . , m}.
such that d i (T ) is the number of times i appears in T . Since Schur polynomials are homogeneous polynomials, Newt(s λ (x 1 , . . . , x m )) is an (m − 1)-dimensional polytope in R m . Consequently, the polytopes for the Schur polynomials in Example 6 and Example 7 are 2-dimensional polytopes in R 3 . In Figure 1, we have depicted these polytopes (equivalently) in the plane for convenience. It is known that Schur polynomials have SNP [19, Proposition 2.5]. Figure 1: Newton polytopes for Schur polynomials in Example 6 (left) and Example 7 (right) drawn in R 2 rather than in a 2-dimensional subspace of R 3 . Both of these polytopes can be shown to be reflexive if we translate them so that their unique interior point is (0, 0).
Symmetric Grothendieck polynomials can be thought of as an inhomogeneous analogue of Schur polynomials. The following definition is due to Lenart [18, Theorem 2.2].
Definition 9. Let x = (x 1 , . . . , x m ) and let λ be a partition with at most m parts. For any partition µ ⊇ λ with at most m rows, let a λµ be the number of fillings of the skew shape µ/λ such that the filling increases strictly along each row and each column, and the filling in the r-th row is from {1, . . . , r − 1}. Let The symmetric Grothendieck polynomial indexed by λ is Example 10. Let λ = (2, 1, 0) 3, m = 3, and x = (x 1 , x 2 , x 3 ). Then See Figure 3 for an illustration of the Newton polytope of G (2,1,0) (x).
Escobar and Yong [12] have shown that symmetric Grothendieck polynomials G λ (x) have SNP.

The Integer Decomposition Property
In this section we will show that the Integer Decomposition Property (IDP) holds for Schur polynomials and a generalization of the symmetric Grothendieck polynomials.

The Newton polytope of a Schur polynomial
Using the realization of the Newton polytope Newt(s λ (x)) as the λ-permutohedron P m λ , we show that all Newton polytopes of Schur polynomials have IDP. One should note that this result is already known by the theory of generalized permutohedra and polymatroids (see, e.g., [24,Corollary 46.2c]). However, we provide our proof as it motivates our methods for the Newton polytope of symmetric Grothendieck polynomials.
Proposition 11. Let λ be a partition with at most m parts and let x = (x 1 , . . . , x m ). Then the Newton polytope Newt(s λ (x)) = P m λ has the integer decomposition property.
Proof. The vertices of the t-th dilate tP m λ are the vertices of P m λ scaled by t, so the vertices of tP m λ are given by the S m -orbit of tλ, and tP m λ = Newt(s tλ (x)). Let p be a point in the t-th dilate tP m λ = Newt(s tλ (x)). Since s tλ (x) has saturated Newton polytope, then p is the content vector of a semistandard Young tableaux T of shape tλ. The tableau T decomposes into t semistandard Young tableaux T 1 , . . . , T t each of shape λ such that T i consists of the j-th columns of T for j ≡ i mod t. Letting v i denote the content vector of T i , then p = v 1 + · · · + v t , so Newt(s λ (x)) has IDP.
Example 12. Let m = 3, x = (x 1 , x 2 , x 3 ), and λ = (2, 1, 0) 3. Each lattice point in the dilated polytope 3Newt(s λ (x)) = Newt(s 3λ (x)) is the content vector of a semistandard Young tableau T of shape 3λ = (6, 3, 0), and the lattice point can be decomposed into the sum of three points which are content vectors of semistandard Young tableaux T 1 , T 2 , T 3 of shape λ by taking the columns of T mod 3.

The Newton polytope of a symmetric Grothendieck polynomial
A notable difference between the Newton polytope of Schur polynomials versus symmetric Grothendieck polynomials is that unlike the case of Schur polynomials, tNewt(G λ (x)) = Newt(G tλ (x)). Motivated by our study of the integer decomposition property of the Newton polytope of symmetric Grothendieck polynomials, we make the following definition.
Definition 13. Let h be a positive integer. Let x = (x 1 , . . . , x m ) and let λ n be a partition with at most m parts. For any partition µ ⊇ λ with at most m rows, let b h,λµ be the number of fillings of the skew shape µ/λ such that the filling increases strictly along each row and each column, and the filling in the r-th row is from {1, . . . , h(r − 1)}. Let the electronic journal of combinatorics 27 (2020), #P00 The inflated symmetric Grothendieck polynomial indexed by λ and h is Example 14. Let λ = (2, 1, 0) 3, m = 3, h = 2, and x = (x 1 , x 2 , x 3 ). Then Compare with Example 10.
Escobar and Yong [12] showed that the symmetric Grothendieck polynomial G λ (x) has SNP and described the components of the Newton polytope associated to the homogeneous components of G λ (x). We extend the work of Escobar and Yong to G h,λ (x) and show that G h,λ (x) also has SNP.

Definition 17.
Let h be a positive integer and let λ be a partition with at most m parts. Let λ (0) = λ and for k 1, let λ (k) |λ| + k be the partition obtained by adding a box to the r k -th row of λ (k−1) , where r k ∈ [m] is the smallest integer such that and adding a box to the r k -th row of λ (k−1) results in a valid partition. If deg G h,λ (x) = |λ| + N , we say λ (0) , . . . , λ (N ) is the sequence of dominating partitions for G h,λ (x).
We justify this terminology with the next result. Lemma 18(a) is an extension of the result [12, Claim A] of Escobar-Yong to the case of inflated symmetric Grothendieck polynomials.
the electronic journal of combinatorics 27 (2020), #P00 Proof. Let µ ∈ A(h, λ) such that µ |λ| + k. Suppose for contradiction that λ (k) does not dominate µ, so that there exists a minimum s > 1 such that µ 1 +· · ·+µ s−1 λ s . This implies µ s > λ (k) s . The partition λ (k) was obtained by adding a box to λ (k−1) in the r k -th row. If s < r k , then , so a box would have been added to λ (k−1) in the s-th row to obtain λ (k) , contradicting the construction of λ (k) . Thus s r k . But then by the construction of λ (k) , for all j r k , which contradicts the existence of s, so part (a) holds.
Parts (b) and (c) follow from the maximality of λ (N ) .

Proposition 19.
Let h be a positive integer, and let λ be a partition with at most m parts. Suppose deg G h,λ (x) = |λ| + N , and let λ (0) , . . . , λ (N ) be the sequence of dominating partitions for G h,λ (x). Further, let H k be the hyperplane in R m defined by
Proof. In [12], the proof that G 1,λ (x) = G λ (x) has SNP does not depend on the inflation parameter h other than in [12, Claim A], which describes the structure of Newt(G λ (x)) arising from the homogeneous components of G λ (x). Using the description of the homogeneous components of G h,λ (x) from Proposition 19, the rest of the proof in [12] applies to arbitrary h ∈ Z 1 and shows that G h,λ (x) has SNP.

Inflated symmetric Grothendieck polynomials and IDP
We now show that the Newton polytopes of inflated symmetric Grothendieck polynomials have IDP. As a corollary, symmetric Grothendieck polynomials have IDP.
By Proposition 19, we know that Newt(G h,λ (x)) is the convex hull of the S m -orbits of the sequence of dominating partitions λ (0) , . . . , λ (N ) , but the next result shows that it suffices to take a certain subset of these partitions. To prove Proposition 24 we will need the following definition.
By the Minkowski-Weyl Theorem a polytope is the convex hull of the set of its extreme points (vertices). Thus, Newt(G h,λ (x)) is the convex hull of its extreme points.
As λ (N ) = (λ 1 + a 1 , . . . , λ m + a m ), then the largest number of boxes that can be added to the r-th row of λ is a r . Thus λ r λ (k) r λ r +a r for each k = 0, . . . , N . By construction, for each i = 1, . . . , m, so that each part of the partition is either at a maximum or a minimum. Thus if µ, ν ∈ Newt(G h,λ (x)) are lattice points such that tµ On the other hand, suppose k / ∈ {b 1 , . . . , b m }. Then there exists j such that with 0 < c < a j . In this case, we have Lastly, a 1 = 0 and N = a 1 + · · · + a m , so In Figure 3, we see that that λ (2) = (2, 2, 1) is not an extreme point of the Newton polytope.
Recall that for a symmetric Grothendieck polynomial G λ (x) we saw that tNewt(G λ (x)) = Newt(G tλ (x)). Inflated symmetric Grothendieck polynomials are defined to address this discrepancy.
If λ has a maximum of a r addable boxes in its r-th row, then at least ta r boxes can be added to the r-th row of tλ, and so ta r a r for r = 1, . . . , m.
Suppose ta r < a r th(r − 1). Then a r < h(r − 1) implies that the (r − 1)-th and r-th rows of λ (N ) have the same length. In other words, λ r−1 + a r−1 = λ r + a r . But by the induction hypothesis, which contradicts the fact that (tλ) (N ) is a partition. Therefore, ta r = a r for r = 1, . . . , m.
Theorem 27. Let λ be a partition with at most m parts and let x = (x 1 , . . . , x m ). Then the Newton polytope Newt(G h,λ (x)) has the integer decomposition property.
Let ν = tλ and let p be a lattice point in the t-th dilate tP = Q. The polynomial G th,tλ (x) has SNP, and by Proposition 24, the point p is a lattice point in Newt(s ν (k) ) for some k, so p is the content vector of a semistandard Young tableau T of shape ν (k) ∈ A(th, tλ). The tableau T decomposes into t semistandard Young tableaux T 1 , . . . , T t , where the tableau T i of shape θ(i) is obtained by taking the j-th columns of T for j ≡ i mod t. We shall show that the partitions θ(1), . . . , The tableau T i is comprised of every t-th column of T , so its shape is Corollary 28. The Newton polytope Newt(G λ (x)) has the integer decomposition property.

Reflexive and Gorenstein Newton polytopes of Schur polynomials
We wish to characterize the Newton polytopes arising from Schur polynomials that are reflexive. These polytopes are (m − 1)-dimensional polytopes in R m ; they are contained in the hyperplane m i=1 x i = m. If we project onto the first m − 1 coordinates, and then translate by (−1, −1, . . . , −1), we get an (m − 1)-dimensional polytope in R m−1 , with 0 in its interior. So we can apply the equivalent condition for reflexivity observed in Section 2.1 directly to the Newton polytope in R m . That is, to show the Newton polytope is reflexive we show there is a unique lattice point in the relative interior and it is lattice distance 1 from each facet. Thus, to classify the reflexive λ-permutohedra, we first identify the facet-defining hyperplanes.
In this section we will often abuse terminology by using "interior" to mean "relative interior" of a polytope, when the affine span of the polytope is clear.
Let λ be a partition with at most m parts and let x = (x 1 , . . . , x m ). Recall that the Newton polytope Newt(s λ (x)) is the λ-permutohedron P m λ , which is the convex hull of the S m -orbit of (λ 1 , . . . , λ m ) in R m . This polytope is of dimension m − 1, and is determined by Rado's inequalities [22]: Note that whether one of Rado's inequalities is facet-defining depends only on |I| and λ, and not on the set I itself.
Theorem 29. The facets of P m λ are determined by the following inequalities.
First we consider the following special case when m < |λ|.
Proof. The λ-permutohedron P m λ is the convex hull of the S m -orbit of (m, . . . , m, 0) in R m . The polytope lies in the hyperplane m i=1 x i = (m − 1)m, and by Theorem 29 the facets are determined by the inequalities x i m for i = 1, . . . , m. Thus, the lattice point (m − 1, . . . , m − 1) in the interior of P m λ is unique, and it is lattice distance 1 from the facets. Therefore, P m λ is reflexive.
We next consider the case m = |λ|.
Proof. We shall show in each case that (1, . . . , 1) is the unique interior lattice point in P m λ and it is lattice distance one from every facet of the Newton polytope.
(a) The polytope P m λ is the convex hull of the X m -orbit of (m, 0, . . . , 0) in R m . By Theorem 29, the facet-defining inequalities of P m λ are just those for |I| = m − 1, and these can be written as is the unique lattice point in the interior of P m λ , and is lattice distance 1 from all facet-defining hyperplanes.
Therefore we conclude in each case that P m λ is reflexive.
We next prove that Propositions 30 and 31 give a complete list of reflexive permutohedra. To do this, we analyze the unique interior point of the reflexive polytope up to translation. Let P • denote the (relative) interior of the polytope P.
Lemma 32. Let m 2 and let λ n be a partition with at most m parts. If |(P m λ ) • ∩ Z m | = 1, then m|n.
Proof. If a lattice point is contained in the interior of P m λ then so is its entire S m -orbit. Thus, the only candidate for a single interior point is ( n m , . . . , n m ) which is only a lattice point when m|n. Proof. The vertex description of λ-permutohedra implies that P m λ is precisely the polytope P m λ translated linearly by the vector (λ m , . . . , λ m ).
In this case, we say that λ reduces by translation to λ .
Theorem 34. Let m 2 and let λ = (λ 1 , . . . , λ m ) n be a partition with at most m parts. The Newton polytope Newt(s λ (x)) = P m λ is reflexive if and only if λ reduces by translation to λ of the following form: Proof. Propositions 30 and 31, and Lemmas 32 and 33 show that if λ reduces to one of these forms, then P m λ is reflexive. We now prove the converse. Suppose P m λ is reflexive. By Lemma 32 we may assume that m|n and P m λ has the unique interior lattice point ( n m , . . . , n m ). By Lemma 33, if (λ) = m, we may replace λ by its translation by (−λ m , . . . , −λ m ). Thus we assume that λ m = 0.
The interior lattice point (1, . . . , 1) is lattice distance k from this facet, so P m λ is reflexive implies k = 1, giving the second case on the list of possible λ .
Lastly, if m − 2k = 0 or 1, this gives the remaining cases on the list of possible λ . Thus this completes the proof that P m λ is reflexive implies that λ reduces to one of the λ on the list.
The result of Theorem 34 allow us to give a characterization of the Gorenstein property as a corollary.
Corollary 35. Let λ be a partition with at most m parts. The Newton polytope P m λ = Newt(s λ (x)) is Gorenstein if and only if P m λ is reflexive or λ reduces by translation to λ of the following form: Proof. If P m λ is Gorenstein, there exists a ∈ Z 1 so that aP m λ = P m aλ is reflexive. If a = 1, then P m λ is reflexive. Assume P m aλ is reflexive for some a 2. Then (aλ) = aλ is one of the cases in Theorem 34.

Reflexive Newton polytopes of inflated symmetric Grothendieck polynomials
We begin by determining the set of facet-defining hyperplanes of the Newton polytope of an inflated symmetric Grothendieck polynomial. From this, we deduce which Newton polytopes are reflexive.
Definition 37. Let f be a linear functional and let H = {x ∈ R m | f (x) = a} be an affine hyperplane in R m . Define the closed half-spaces For any set S ⊆ R m , we say H isolates S if S is contained in H + or H − .
Given an m-dimensional polytope P, if H is an affine hyperplane which isolates P and dim(P ∩ H) = m − 1, then H is a facet-defining hyperplane of P.
We pinpoint some facet-defining inequalities of Newt(G h,λ (x)) to show that if it is reflexive, then there is a very limited region where its unique interior lattice point may lie.
Let F = F (I) be a facet of Newt(s λ (b k ) (x)) defined by the inequality (Recall from Theorem 29 that all facet-defining inequalities of Newt(s λ (b k ) (x)) are of this form.) Since Newt(s λ (b k ) (x)) lies in the affine hyperplane x 1 + · · · + x m = |λ (b k ) |, then F is of the form and the affine hyperplanes are given by We will determine which of these hyperplanes isolate Newt(G h,λ (x)).

Lemma 40. Let I be a proper nonempty subset of [m], let F = F (I) be a facet of
= L for some k = 1, . . . , m, and let H(F ) 1 , H(F ) 0 be the hyperplanes associated to F . Then H(F ) 1 isolates Newt(G h,λ (x)) if |I| k, and H(F ) 0 isolates Newt(G h,λ (x)) if |I| k.
Proof. As each Newt(s λ (b ) (x)) is the convex hull of the S m -orbit of λ (b ) , it suffices to show that the S m -orbits of λ (b ) for = 1, . . . , m all lie on one side of the proposed Newt(G h,λ (x))-isolating hyperplane.
Let π ∈ S m , so that π(λ (b ) ) = (λ As the partition λ (b k ) is obtained by adding the maximum allowable number (a ) of boxes to the -th row of λ for = 1, . . . k, then for i k, And for i > k, If |I| k, then H(F ) 1 isolates Newt(s λ (b ) (x)) for all = 0, . . . , N since In other words, and Newt(G h,λ (x)) ⊆ (H(F ) 0 ) + . The result now follows.
We will see that nearly all of the Newt(G h,λ (x))-isolating hyperplanes of Lemma 40 are facet-defining. We first identify some that are not.
We can further narrow down the set of Newt(G h,λ (x))-isolating hyperplanes of Lemma 40 that are facet-defining.
Proof. Recall that λ (bm) = λ (N ) and it is distinct from λ (b m−1 ) since we assumed that λ m = 0. The only difference between the partitions λ (b m−1 ) and λ (bm) is in their mth row, so no additional Newt(G h,λ (x))-isolating hyperplane that arises from a facet of Newt(s λ (bm) (x)) is introduced.
= L for some k = 1, . . . , m. Suppose H(F ) is a hyperplane associated to F which isolates Newt(G h,λ (x)) and is not of the form J j . Then H(F ) is a facet-defining hyperplane of Newt(G h,λ (x)).
Second, suppose H(F ) is of the form H(F ) 0 so that by Lemma 40, Newt(G h,λ (x)) lies in the half-space with |I| k. Furthermore by Lemma 42, it suffices to assume k = 2, . . . , m. So if π ∈ S m is a permutation such that π −1 (I) = {1, . . . , |I|}, then noting that the partition λ (b k −1) exists since k 2. Thus π(λ (b k −1) ) is a point that lies on In both of these cases, note that dim( Proof. If Newt(G h,λ (x)) is reflexive, then by Corollary 39 its unique interior lattice point u lies on Newt(s λ (1) (x)), so u is also the unique interior lattice point of Newt(s λ (1) (x)).
Suppose H is an affine hyperplane in R m such that H is a facet-defining hyperplane of Newt(s λ (1) (x)). The partitions λ (0) and λ (1) differ by exactly one box, and if this occurs in the r-th row, then λ (0) ⊂ λ (1) ⊆ λ (br) for some r 2. By the arguments in Theorem 43, if Newt(s λ (1) (x)) ⊆ H − , then H is a facet-defining hyperplane of Newt(s λ (br ) (x)), and if Newt(s λ (1) (x)) ⊆ H + , then H is a facet-defining hyperplane of Newt(s λ (0) (x)). Moreover, Theorem 43 states that in either case, H is a facet-defining hyperplane of Newt(G h,λ (x)), and since it is reflexive, then u is lattice distance one from H. Thus it follows that Newt(s λ (1) (x)) is reflexive.
It remains to prove that the facet-defining inequalities of Newt(G h,λ (x)) are precisely the ones appearing in Lemma 38 and Theorem 43.
Theorem 47. The facets of Newt(G h,λ (x)) are determined by the following inequalities.
, if I is a nonempty proper subset of [m], |I| k, and the inequality is a facet-defining inequality of Newt(s λ (b k ) (x)).
, if I is a nonempty proper subset of [m], |I| k, and the inequality is a facet-defining inequality of Newt(s λ (b k ) (x)).
Proof. Suppose H is a facet-defining hyperplane of Newt(G h,λ (x)). Then there exists a k such that H ∩ Newt(s λ ) and by Rado's inequalities, H is defined by i∈I x i = a for some a > 0 and I ⊂ [m]. Recall the electronic journal of combinatorics 27 (2020), #P00 Suppose H is a facet-defining hyperplane of Newt(G h,λ (x)) so that, in particular, for k = 1, . . . , m.
Since the symmetric group S m acts on Newt(G h,λ (x)) (and hence its facet-defining hyperplanes) then without loss of generality, it suffices to assume that H is defined by an equation of the form r i=1 x i = d for some r m. First assume that Newt(G h,λ (x)) ⊆ H − . Since H is facet-defining, then it contains at least one vertex of Newt(G h,λ (x)), say λ (b k ) , where we choose k to be the smallest index for which this is true. By the same argument as in the proof of Lemma 42, we may assume that k m − 1.
for all i and all k, then where the second inequality is due to Newt(G h,λ (x)) ⊆ H − . So the vertices λ (b ) also lie in H for all k.
From this observation, we see that the vertices of Newt(G h,λ (x)) that lie in H consists of concatenating an S r -permutation of (λ m . Let s be the dimension of the convex hull of the permutations of (λ But H is a facet-defining hyperplane of Newt(G h,λ (x)), so s + m − r = m − 1 implies s = r − 1. Now, the dimension of H ∩ Newt(s λ (b k ) (x)) is the sum of s and the dimension of the convex hull of the permutations of (λ the electronic journal of combinatorics 27 (2020), #P00 Thus H is a facet-defining hyperplane of Newt(s λ (b k ) (x)).
A similar argument works in the case Newt(G h,λ (x)) ⊆ H + . Therefore the result follows.
We are now ready to classify the Newton polytopes of inflated symmetric Grothendieck polynomials that are reflexive. First, we examine the simple case when m = 2.
Proposition 49. Let m 3 and assume that λ is reduced by translation so that λ = (λ 1 , . . . , λ m−1 , 0). If Newt(G h,λ (x)) is reflexive, then λ is of one of the following forms: Since λ is assumed to be reduced by translation, then c = 1 and λ = (m + 1, 1, . . . , 1, 0) or c = 0 and λ = (m − 1, 0, . . . , 0). In the first case, this implies that the box that was added to λ to obtain λ (1) is in the m-th row (where m 3), but by definition it should have been in the second row so this case is not possible. In the second case, this implies that the box that was added to λ to obtain λ (1) is in the first row but this is also not possible by definition.
The result now follows.
We next determine the values of h for which the partitions λ in Proposition 49 give rise to reflexive Newton polytopes of inflated symmetric Grothendieck polynomials.
Proposition 51. Let m 3 and let Then Newt(G h,λ (x)) is reflexive for h 1.
Evidently, u is lattice distance one from every facet-defining hyperplane of the Newton polytope, so Newt(G h,λ (x)) is reflexive.
By Theorem 47, the facet-defining hyperplanes of Newt(G h,λ (x)) are x i = 0, 2 for i = 1, . . . , m, and x 1 + · · · + x m = m − 1, and we see that Newt(G h,λ (x)) is the truncation of the m-cube [0, 2] m by the hyperplane x 1 + · · · + x m = m − 1. The unique interior lattice point u = (1, . . . , 1) is lattice distance one from each of these hyperplanes, and we conclude that Newt(G h,λ (x)) is reflexive. Tables of h * -vectors for reflexive polytopes in the family Newt(G h,λ (x)) can be found in the Appendix.

h * -polynomials
Thus far we have shown that the Newton polytopes arising from Schur polynomials have the integer decomposition property (Proposition 11). In addition, we have classified the polytopes that are reflexive (Theorem 34) and Gorenstein (Corollary 35). Following the motivation of Conjecture 1, we study the h * -polynomials of these polytopes.
In this section, we provide closed-form expressions for the h * -polynomials of the four families of reflexive Newton polytopes of Schur polynomials from Theorem 34. We also prove that all of these h * -polynomials are unimodal. Figure 4 depicts examples of reflexive and nonreflexive Newt(s λ (x)) with their corresponding h * -vectors. The h * -vectors of reflexive Newt(s λ (x)) for several partitions are provided in Table 3 in the Appendix. Figure 4: The reflexive polytope Newt(s (2,2,0,0) (x)) has h * -vector (1,15,15,1) and is shown on the left. The polytope Newt(s (3,1,0,0) (x)), which has h * -vector (1, 27, 31, 1), is shown on the right and is not reflexive.
Proposition 53. Let λ = (n). The h * -polynomial of the Newton polytope Newt(s λ (x)) = P n λ is of degree n − 1, and has coefficients Proof. Let P = P n λ . We first calculate the Ehrhart polynomial ehr P (k). The polytope P has vertices ne i for 1 i n and the polytope kP has vertices kne i for 1 i n, where e i is the i th standard basis vector. The lattice points in kP are all the points (v 1 , v 2 , . . . , v n ) with all v i nonnegative integers and i v i = kn. These are in natural bijection with all weak compositions of kn into n parts. The number of these is well known, so we get ehr P (k) = kn + n − 1 n − 1 .
Thus the Ehrhart series of P is To get the h * -polynomial, we multiply the Ehrhart series by (1 − x) n .
So for 0 j n − 1, Remark 54. The coefficients in Proposition 53 are precisely those given by OEIS Sequence A108267 [25].
Proof. Let Q be the translation of P n λ so that the origin is its unique interior lattice point. Then Q is the convex hull of all vectors e i − e j for all i, j ∈ [n]. Thus Q is precisely the polytope P A n−1 as stated in [2,Theorem 2]. This theorem shows that h * j = n−1 j 2 .
Proof. We proceed in the same way as Proposition 53. Let n be even and λ = (2, . . . , 2) n. Define P = P n λ . We first calculate the Ehrhart polynomial ehr P (k). The vertices of P are 2 i∈I e i for all I ⊆ [n] such that |I| = n/2. Thus kP has vertices 2k i∈I e i for these same I. Therefore the lattice points in kP are the points (v 1 , . . . , v n ) such that 0 v i 2k and i v i = kn. There are in natural bijection with the weak compositions of kn into n parts such that each part has size less than or equal to 2k. This number is counted by a n,k above. Therefore ehr P (k) = a n,k .
Thus the Ehrhart series of P is ∞ k=0 ehr P (k)x k = ∞ k=0 a n,k x k .
To get the h * -polynomial, we multiply the Ehrhart series by (1 − x) n : a n,k x k So for 0 j n − 1, the coefficient of x j in the above series is h * j = j k=0 (−1) j−k n j − k a n,k .
Now suppose that n is odd and λ = (2, . . . , 2, 1) n. Again define P = P n λ . The vertices of P are e j + 2 i∈I e i for all j / ∈ I ⊆ [n] with |I| = n/2 . Thus kP has vertices k(e j + 2 i∈I e i ) for these same I and j. Therefore the lattice points in kP are the points (v 1 , . . . , v n ) such that 0 v i 2k and i v i = kn. Thus the h * -polynomial calculation in the even case works precisely the same way in the odd case. Now that we have determined the h * -polynomials for these reflexive polytopes, we verify that Conjecture 1 holds in the case of Newton polytopes of Schur polynomials. That is, we show that the coefficients of the h * -polynomials of the polytopes listed in Theorem 34 are unimodal.
Proposition 57. All Newton polytopes arising from Schur polynomials that are reflexive have h * -polynomials with unimodal coefficients.
Proof. We note that if P is reflexive and P has a regular unimodular triangulation, then the h * -polynomial of P has unimodal coefficients by Theorem 3. Thus, we can prove this result by showing that these polytopes have regular unimodular triangulations.
If P is the polytope from Proposition 53, then P is a dilate of a unimodular simplex. By [14,Theorem 4.8], the dilate of any polytope with a regular unimodular triangulation has a regular unimodular triangulation.
If P is the polytope from Proposition 55, we observe that the coefficients are logconcave by a routine calculation since binomial coefficients are log-concave.
If P is the polytope from Proposition 56 when n is even, then P is a dilate of a hypersimplex, which is known to have a unimodular triangulation by [17].
If P is the polytope from Proposition 56 when n is odd, we note that the facets for this polytope are listed in Proposition 31(c). We can translate P so that the origin is the unique interior point of P. The facet-defining matrix for this translate of P is easily seen to be unimodular, so by [14,Theorem 2.4], P has a regular unimodular triangulation. Thus the conjecture holds for this family of polytopes.
Remark 58. One could also consider the question of h * -unimodality for Gorenstein P m λ , but these results follow rather quickly. Several of the Gorenstein examples are hypersimplices (Corollary 36) which are known to have combinatorial formulas [16] and are known to have regular unimodular triangulations [17]. The remaining examples of Gorenstein P m λ are dilated standard simplices and arguments will follow in a similar vein to the case of λ = (n).

Conclusion
In this paper we study the Newton polytopes of Schur polynomials and show that they all have the integer decomposition property. We determine which Schur polynomials have reflexive Newton polytopes, and for which the Newton polytope is Gorenstein. For the reflexive Newton polytopes of Schur polynomials, we give the h * -polynomials, and show that they are unimodal.
We also introduce a generalization of symmetric Grothendieck polynomials, called inflated symmetric Grothendieck polynomials. We show that all these polynomials have saturated Newton polytope and their Newton polytopes have the integer decomposition property. We characterize the partitions whose inflated symmetric Grothendieck polynomials have reflexive Newton polytopes and provide a table of their h * -vectors.
The study of polynomials with saturated Newton polytope both introduces additional lattice polytopes of combinatorial interest and provides a new tool for approaching problems in Ehrhart theory. It may be fruitful to consider questions of the integer decomposition and reflexive polytopes for the Newton polytopes of other families of polynomials which are known to have SNP, such as chromatic symmetric polynomials and Schubert polynomials. Additionally, it is worth noting that the Newton polytopes of many polynomials of interest in algebraic combinatorics appear not only to have SNP, but also to have nice Ehrhart theoretic properties such as IDP and h * -unimodality, either by our theorems or computationally. Perhaps it would be of interest to investigate if these Newton polytopes have other sought after properties, such as Ehrhart positivity.