Newton polytope of good symmetric polynomials

We introduce a general class of symmetric polynomials that have saturated Newton polytope and their Newton polytope has integer decomposition property. The class covers numerous previously studied symmetric polynomials.


Introduction
In combinatorics, if a convex polytope equals the convex hull of its integer points, we say that it is a lattice polytope.Studying lattice polytopes is important because of their connections in many other domains.For instance, in mathematical optimization, if a system of linear inequalities defines a polytope, then we can use linear programming to solve integer programming problems for this system (see [Bar17]).In algebraic geometry, lattice polytopes are used to study projective toric varieties (see [CLS11,Ful16]).The Newton polytope is a lattice polytope associated with a polynomial: it is the convex hull of exponent vectors.The Newton polytope is a central object in tropical geometry (see [KKE21]), and they are used to characterizing Grobner bases (see [Stu96]).
Lattice polytopes are studied by Ehrhart polynomials (see [Eug62]).Important properties of Ehrhart polynomials such as unimodality and log-concavity are related to the integer decomposition property (IDP) of the lattice polytope (see [OH06,BR07,SVL13]).In [BGH + 21], the authors studied the Newton polytope of inflated symmetric Grothendieck polynomials.The saturated property (SNP) of inflated symmetric Grothendieck polynomials in [BGH + 21] generalizes the SNP of symmetric Grothendieck polynomials in [EY17].The SNP of the inflated symmetric Grothendieck polynomials is an important point to derive the IDP of their Newton polytope.
in Representation Theory", 2022-2023 (Decision No.1323/QD-BGDDT, May 19, 2022).Khanh was partially supported by the NSF grant DMS-1855592 of Prof. Cristian Lenart.Hiep would like to thank Vietnam Institute for advanced study in Mathematics for the very kind support and hospitality during his visit.We are grateful to the referee for valuable comments to improve the text.

Newton polytope
Example 2.1.The convex hull P of twelve points in R 3 below is a lattice polytope.
The permutations of (3, 1, 0) are vertices of the polytope P. In the picture below, P is the blue hexagon.(1, 0, 3) (0, 1, 3) (0, 3, 1) Let P be a lattice polytope.For a positive integer t, let tP = {tv | v ∈ P}.We say that P has integer decomposition property (IDP) if, for any positive integer t and p ∈ tP ∩ Z m , there are
We see that (9, 2, 1) ∈ 3P ∩ Z 3 and is the sum of three points in P ∩ Z 3 .
We have (1, 1, 1) ∈ 2G ∩ Z 3 , but it can not be written as a sum of two points in G ∩ Z 3 .So G does not have IDP.
We say that f has satured Newton polytope (SNP

Schur polynomials
A partition with at most m parts is a sequence of weakly decreasing nonnegative integers λ = (λ 1 , . . ., λ m ).The size of partition λ is defined by |λ| = m i=1 λ i .Each partition λ is presented by a Young diagram Y (λ) that is a collection of boxes such that the leftmost boxes of each row are in a column, and the numbers of boxes from the top row to bottom row are λ 1 , λ 2 , . . ., respectively.A semistandard Young tableau of shape λ with entries from {1, . . ., m} is a filling of the Young diagram Y (λ) by the ordered alphabet {1 < • • • < m} such that the entries in each column are strictly increasing from top to bottom, and the entries in each row are weakly increasing from left to right.A Young tableau T is said to have content α = (α 1 , α 2 , . . . ) if α i is the number of entries i in the tableau T .We write x T = x α = x α 1 1 x α 2 2 . . . .For each partition λ with at most m parts, the Schur polynomial s λ (x 1 , . . ., x m ) is defined as the sum of x T , where T runs over the semistandard Young tableaux of shape λ with filling from {1, . . ., m}.

Good symmetric polynomials
Let α and β be partitions with at most m parts.We say β is bigger than α and write β ≥ α if and only if β i ≥ α i for all i.If α, β are partitions of the same size, we say β dominates α and write β α if Let F (x 1 , . . ., x m ) be a linear combination of Schur polynomials associated to partitions with at most m parts.We can collect Schur polynomials appearing in F associated with partitions of the same size to a bracket.We say that F is good if it satisfies the following conditions: (a) The support of each bracket equals the union of supports of its Schur elements.
(b) Suppose that there are l + 1 brackets in condition (a).In each bracket, there is a unique -maximum partition.These -maximum partitions have a form where α ≤ β are fixed partitions and for each i > 0, λ i is obtained from λ i−1 by adding a box in the northmost row of λ i−1 such that the addition gives a Young diagram, α < λ i ≤ β.Then F is a good polynomial.In particular, F has SNP and N ewton(F ) has IDP.
Example 4.4.Let F (x 1 , x 2 , x 3 ) be Schur polynomials in the same bracket have the same sign.The -maximum partitions λ i for i = 0, . . ., 5 chosen from brackets have form 3. For a positive integer t, we construct a chain of form (1) Then F t is a good linear combination of Schur polynomials and Λ (i) = tλ (i) for each i = 0, . . ., m.By (10), we have 4. Let p a point in tN ewton(F ) ∩ Z m .By (12), p is a point in N ewton(F t ) ∩ Z. Since F t has SNP, by (9), it is a point in N ewton(s Λ i ) ∩ Z for some Λ i in (11).Hence, p is the content of some semistandard tableau T of shape Λ i with filling from {1, . . ., m}.
For j = 1, . . ., t, let T j be the semistandard tableau obtained by taking j ′ -th column of T for j ′ ≡ j mod t.Let θ(j) be the shape of tableau T j .Let v j be the content of tableau T j .Then So by (9), v j is a point of N ewton(F ) ∩ Z m .Therefore we conclude that N ewton(F ) has IDP.

Theorem 4. 2 .
Let F be a good linear combination of Schur polynomials.Then F has SNP and Newton(F) has IDP.Corollary 4.3.Let F be a linear combination of Schur polynomials such that the condition (a) is replaced by (a') or the condition (b) is replaced by (b') below:(a') any two Schur polynomials in the same bracket of F have the same sign, (b') there exists partitions λ, λ so that s µ appears in F if and only if λ ≤ µ ≤ λ.

Hence,F
is a good symmetric polynomial.N ewton(F ) is the convex hull of six different color polygons in the picture below.Each polygon is the Newton polytope of each bracket.In fact, F is the inflated symmetric Grothendieck polynomial G 2,(3,1,0) in [BGH + 21].Hence, F has SNP and N ewton(F ) has IDP by [BGH + 21, Proposition 21, Theorem 27].