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Newton Polytopes in Algebraic Combinatorics Newton Polytopes in Algebraic Combinatorics Neriman Tokcan University of Michigan, Ann Arbor April 6, 2018 Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Outline Newton Polytopes Geometric Operations on Polytopes Saturated Newton Polytope (SNP) The ring of symmetric polynomials SNP of symmetric functions Partitions - Dominance Order Newton polytope of monomial symmetric function SNP and Schur Positivity Some other symmetric polynomials Additional results Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Newton Polytopes d A polytope is a subset of R ; d ≥ 1 that is a convex hull of finite d set of points in R : Convex polytopes are very useful for analyzing and solving polynomial equations. The interplay between polytopes and polynomials can be traced back to the work of Isaac Newton on plane curve singularities. P α The Newton polytope of a polynomial f = n c x α2Z ≥0 α 2 C[x1; :::; xn] is the convex hull of its exponent vectors, i.e., n Newton(f ) = conv(fα : cα 6= 0g) ⊆ R : Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Geometric operations on Polytopes d Let P and Q be two polytopes in R , then their Minkowski sum is given as P + Q = fp + q : p 2 P; q 2 Qg: The geometric operation of taking Minkowski sum of polytopes mirrors the algebraic operation of multiplying polynomials. Properties: Pn Pn n conv( i=1 Si ) = i=1 conv(Si ); for Si ⊆ R : Newton(fg) = Newton(f ) + Newton(g); for f ; g 2 C[x1; :::; xn]: Newton(f + g) = conv(Newton(f ) [ Newton(g)): Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Saturated Newton Polytope{SNP A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. Instances of SNP in algebraic combinatorics are compiled in the paper \Newton Polytopes in Algebraic Combinatorics" (joint work with Cara Monical and Alexander Yong). Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Saturated Newton Polytopes Example:f (x; y) = axy + bx2 + cy 5 + d does not have SNP. Generally, polynomials are not SNP. Worse still, SNP is not preserved by basic polynomial operations. 2 2 Example: f (x1; x2; x3; x4) = x1 + x2x3 + x2x4 + x3x4 is SNP but f is not SNP (it misses x1x2x3x4). Neriman Tokcan Newton Polytopes in Algebraic Combinatorics The ring of symmetric polynomials Consider the ring Z[x1;:::; xn] of polynomials in n independent variables x1;:::; xn with integer coefficients. The symmetric polynomials form a sub-ring Sn Symn = Z[x1;:::; xn] : Symn is a graded ring: we have k Symn = ⊕k≥0Symn k where Symn consist of homogeneous polynomials of degree k with zero polynomial. Let m ≥ n; setting xi = 0 for i ≥ n + 1 defines a surjective homomorphism Symm Symn: Let Sym denote the lim Sym , i.e., the ring of symmetric functions − n in countably many variables. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics SNP of symmetric functions Definition A symmetric polynomial f has SNP if f (x1;:::; xm) has SNP for all m ≥ 1: Q: Do we need check for all m ≥ 1? If f (x1;:::; xm) has SNP, then f (x1;:::; xn) has SNP for any n ≤ m: If f (x1;:::; xm) for m ≥ deg(f ) has SNP, then f (x1;:::; xn) has SNP for all n ≥ m: Proposition (Stability of SNP) Suppose f 2 Sym has finite degree. Then f has SNP if there exists m ≥ deg(f ) such that f (x1;:::; xm) has SNP. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Partitions{Dominance Order A partition is any sequence λ = (λ1; λ2; : : : ; λr ;:::) of non-negative integers in decreasing order: λ1 ≥ λ2 ≥ ::: ≥ λr ::: The length of λ, denoted by `λ, is the number of parts of the sequence. The weight of λ, denoted by jλj, is the sum of the parts, i.e., jλj = λ1 + λ2 + ::: Par(d)= fλ : jλj = dg denotes the set of all partitions of d: Dominance order ≤D on Par(d) is defined by k k X X µ ≤D λ if µi ≤ λi for all k ≥ 1: i=1 i=1 Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Newton polytope of monomial symmetric polynomial n α For each α = (α1; : : : ; αn) 2 N ; we denote by x the monomial α α1 αn x = x1 ::: xn : X α mλ(x1;:::; xn) = x α where the sum is over distinct permutations of λ. The Newton polytope of mλ is the λ-permutahedron, denoted Pλ; n is the convex hull of the Sn orbit of λ 2 R : Pµ ⊆ Pλ () µ ≤D λ. n F mλ 2 Sym is SNP () λ = (1 ): Neriman Tokcan Newton Polytopes in Algebraic Combinatorics λ-permutahedron Let λ = (4; 3; 2; 1); then Newton polytope of mλ is the permutahedron of order 4. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Schur polynomials We can identify a partition with its Young diagram. A semi-standard Young tableau T is a filling of λ with entries from Z≥0 that is weakly increasing along rows and strictly increasing down columns. Let λ = (3; 1) and the content µ = (1; 1; 2; 1), then we can have 3 different corresponding tableau: Then the corresponding Kostka number Kλ,µ = 3: Kλ,µ 6= 0 () µ ≤D λ Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Schur polynomials and SNP The Schur polynomial is X Sλ = Kλ,µmµ: µ Newton(f ) = conv([µ≤D λNewton(mµ)) = conv([µ≤D λPµ) = Pλ: Since Kλ,µ 6= 0; Sλ has SNP. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Linear Combinations of Schur Polynomials and SNP Proposition Suppose f 2 Symn is homogeneous of degree d such that X f = cµSµ: µ2Par(d) Suppose there exist λ with cλ 6= 0 and cµ 6= 0 only if µ ≤D λ. If n < `λ; f = 0: Otherwise, n 1 Newton(f ) = Pλ ⊆ R : 2 If moreover cµ ≥ 0 for all µ, then f has SNP. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Schur positivity does not imply SNP Example (Schur positive without a unique ≤D -maximal term) Let f = S(8;2;2) + S(6;6): It is enough to show that f (x1; x2; x3) is not SNP. Now, m(8;2;2)(x1; x2; x3) and m(6;6)(x1; x2; x3) appear in the monomial expansion of f (x1; x2; x3): However, m(7;4;1)(x1; x2; x3) is not in f (x1; x2; x3) since (7; 4; 1) is not ≤D comparable with (8; 2; 2) nor (6; 6; 0): 1 1 (7; 4; 1) = 2 (8; 2; 2) + 2 (6; 6; 0) 2 Newton(f ); then f is not SNP. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Schur positivity SNP The Schur positivity assumption in the previous Proposition is necessary. Example 3 3 Let f = S(3;1)(x1; x2) − S(2;2)(x1; x2) = x1 x2 + x1x2 is not SNP. It 2 2 is missig x1 x2 : Having a unique ≤D -maximal term is not necessary. Example f = S(2;2;2) + S(3;1;1;1) is SNP, but (2; 2; 2) and (3; 1; 1; 1) are ≤D incomparable. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Products of Schur polynomials are SNP Proposition 1 2 N Sλ1 Sλ2 ::: SλN has SNP for any partitions λ ; λ ; : : : ; λ : Proof. We have X v SλSµ = cλ,µSv 2 Sym v v where cλ,µ 2 Z≥0 is the Littlewood-Richardson coefficient. (We v will omit the details here). By homogeneity, cλ,µ = 0 unless jvj = jλj + jµj: It can be shown that for λ + µ = (λ1 + µ1; λ2 + µ2;:::): Sλ+µ is the unique ≤D maximal term in the Schur expansion of SλSµ; therefore, the product is SNP. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Involutive automorphism and SNP Let w : Sym ! Sym be the involutive automorphism defined by w(Sλ) = Sλ0 ; where λ0 is the shape obtained by transposing the Young diagram λ. Example (w does not preserve SNP) We showed that f = S(8;2;2) + S(6;6) is not SNP. Now w(f ) = S(3;3;1;1;1;1;1;1) + S(2;2;2;2;2;2) 2 Sym: To see that w(f ) has SNP, it suffices to show that any partition v that is a linear combination of rearrangements of λ = (3; 3; 1; 1; 1; 1; 1; 1) and µ = (2; 2; 2; 2; 2; 2) satisfies v ≤D λ or v ≤D µ. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Forgotten symmetric polynomial The forgotten symmetric functions are defined by fλ = λw(mλ) jλ|−`(λ) where λ = (−1) : Proposition n fλ 2 Sym has SNP if and only if λ = (1 ): Proof. n ((=:) If λ = (1 ); then mλ = S(1n); and fλ = S(n;0;0;:::;0) which is SNP. (=):) X fλ = aλµmµ; aλµ ≥ 0: µ n n It can be shown that if λ 6= (1 ); then aλ1n = 0: Also (1 ) 2 Pµ for n n all µ ` n: Then, (1 ) 2 Newton(fλ). If λ 6= (1 ); f does not have SNP. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Elementary and complete homogeneous symmetric polynomials The elementary symmetric polynomial is defined by X ek (x1;:::; xn) = xi1 xi2 ::: xin 1≤i1<i2<:::<ik ≤n and eλ = eλ1 eλ2 ::: The complete symmetric polynomial hk (x1;:::; xn) is the sum of all degree k homogeneous polynomials and hλ = hλ1 hλ2 :::: Since ek = S(1k ) and hk = S(k); then eλ and hλ can be considered as product of Schur functions. Therefore, they have SNP. Neriman Tokcan Newton Polytopes in Algebraic Combinatorics e-positivity does not imply SNP P f 2 Sym is e-positive if f = λ aλeλ where aλ ≥ 0 for every λ.
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