Inequalities for Symmetric Polynomials

Inequalities for Symmetric Polynomials Curtis Greene October 24, 2009

Co-authors This talk is based on Inequalities for Symmetric Means, with Allison Cuttler, Mark Skandera (to appear in European Jour. Combinatorics). Inequalities for Symmetric Functions of Degree 3, with Jeffrey Kroll, Jonathan Lima, Mark Skandera, and Rengyi Xu (to appear). Other work in progress. Available on request, or at www.haverford.edu/math/cgreene.

Inequalities for Averages and Means Classical examples (e.g., Hardy-Littlewood-Polya) THE AGM INEQUALITY: x 1 + x 2 + + x n n (x 1 x 2 x n ) 1/n x 0.

Inequalities for Averages and Means Classical examples (e.g., Hardy-Littlewood-Polya) THE AGM INEQUALITY: x 1 + x 2 + + x n n (x 1 x 2 x n ) 1/n x 0. NEWTON S INEQUALITIES: e k (x) e k (x) e k (1) e k (1) e k 1(x) e k+1 (x) e k 1 (1) e k+1 (1) x 0 MUIRHEAD S INEQUALITIES: If λ = µ, then m λ (x) m λ (1) m µ(x) m µ (1) x 0 iff λ µ (majorization).

Inequalities for Averages and Means Other examples: different degrees MACLAURIN S INEQUALITIES: ( e j (x) ) 1/j ( e k (x) ) 1/k if j k, x 0 e j (1) e k (1) SCHLÖMILCH S (POWER SUM) INEQUALITIES: (p j (x) n ) 1/j ( pk (x)) 1/k n if j k, x 0

Inequalities for Averages and Means Some results Muirhead-like theorems (and conjectures) for all of the classical families. A single master theorem that includes many of these. Proofs based on a new (and potentially interesting) kind of positivity.

Inequalities for Averages and Means Definitions We consider two kinds of averages : Term averages: F (x) = 1 f (1) f (x), assuming f has nonnegative integer coefficients. And also Means: F(x) = ( ) 1 1/d f (1) f (x) where f is homogeneous of degree d.

Inequalities for Averages and Means Definitions We consider two kinds of averages : Term averages: F (x) = 1 f (1) f (x), assuming f has nonnegative integer coefficients. And also Means: Example: F(x) = ( ) 1 1/d f (1) f (x) where f is homogeneous of degree d. E k (x) = 1 ( n k)e k (x) E k (x) = (E k (x)) 1/k

Inequalities for Averages and Means Muirhead-like Inequalities: ELEMENTARY: E λ (x) E µ (x), x 0 λ µ. POWER SUM: P λ (x) P µ (x), x 0 λ µ. HOMOGENEOUS: H λ (x) H µ (x), x 0 = λ µ. SCHUR: S λ (x) S µ (x), x 0 = λ µ.

Inequalities for Averages and Means The Majorization Poset P 7 (Governs term-average inequalities for E λ, P λ, H λ, S λ and M λ.)

Normalized Majorization Majorization vs. Normalized Majorization MAJORIZATION: λ µ iff λ 1 + λ i µ 1 + µ i i MAJORIZATION POSET: (P n, ) on partitions λ n. NORMALIZED MAJORIZATION: λ µ iff λ λ µ µ. NORMALIZED MAJORIZATION POSET: Define P = n P n. Then (P, ) = quotient of (P, ) (a preorder) under the relation α β if α β and β α. NOTES: (P, ) is a lattice, but is not locally finite. (P n, ) is not a lattice. (P n, ) embeds in (P, ) as a sublattice and in (P n, ) as a subposet.

Normalized Majorization P n partitions λ with λ n whose parts are relatively prime. Figure: (P 6, ) with an embedding of (P 6, ) shown in blue.

Normalized Majorization Muirhead-like Inequalities for Means ELEMENTARY: E λ (x) E µ (x), x 0 λ µ. POWER SUM: P λ (x) P µ (x), x 0 λ µ. HOMOGENEOUS: H λ (x) H µ (x), x 0 = λ µ. CONJECTURE: the last implication is. What about inequalities for Schur means S λ?

Master Theorem: Double Majorization A Master Theorem for Monomial Means THEOREM/CONJECTURE: M λ (x) M µ (x)iff λ µ. where λ µ is the double majorization order (to be defined shortly).

Master Theorem: Double Majorization A Master Theorem for Monomial Means THEOREM/CONJECTURE: M λ (x) M µ (x)iff λ µ. where λ µ is the double majorization order (to be defined shortly). Generalizes Muirhead s inequality; allows comparison of symmetric polynomials of different degrees.

Master Theorem: Double Majorization The double (normalized) majorization order DEFINITION: λ µ iff λ µ and λ µ, EQUIVALENTLY: λ µ iff λ λ µ µ and λ λ µ µ. DEFINITION: DP = (P, )

Master Theorem: Double Majorization The double (normalized) majorization order DEFINITION: λ µ iff λ µ and λ µ, EQUIVALENTLY: λ µ iff λ λ µ µ and λ λ µ µ. DEFINITION: DP = (P, ) NOTES: The conditions λ µ and λ µ are not equivalent. Example: λ = {2, 2}, µ = {2, 1}. If λ µ and µ λ, then λ = µ; hence DP is a partial order. DP is self-dual and locally finite, but is not locally ranked, and is not a lattice. For all n, (P n, ) embeds isomorphically in DP as a subposet.

Master Theorem: Double Majorization DP 5 Figure: Double majorization poset DP 5 with vertical embeddings of P n, n = 1, 2,..., 5. (Governs inequalities for M λ.)

Master Theorem: Double Majorization DP 6 Figure: Double majorization poset DP 6 with an embedding of P 6 shown in blue.

Master Theorem: Double Majorization Much of the conjecture has been proved: MASTER THEOREM : λ, µ any partitions M λ M µ if and only if λ µ, i.e., λ λ µ µ and λ λ µ µ. PROVED: The only if part. For all λ, µ with λ µ. For λ, µ with λ, µ 6 (DP 6 ). For many other special cases.

Master Theorem: Double Majorization Interesting question The Master Theorem/Conjecture combined with our other results about P λ and E λ imply the following statement: M λ (x) M µ (x) E λ (x) E µ (x) and P λ (x) P µ (x).

Y-Positivity Why are these results true? Y-Positivity ALL of the inequalities in this talk can be established by an argument of the following type: Assuming that F (x) and G(x) are symmetric polynomials, let F (y) and G(y) be obtained from F (x) and G(x) by making the substitution x i = y i + y i+1 + y n, i = 1,..., n. Then F (y) G(y) is a polynomial in y with nonnegative coefficients. Hence F (x) G(x) for all x 0.

Y-Positivity Example: AGM Inequality In[1]:= n 4; LHS Sum x i, i, n n ^n RHS Product x i, i, n Out[2]= Out[3]= In[4]:= Out[4]= In[5]:= Out[5]= 1 x 1 x 2 x 3 x 4 4 256 x 1 x 2 x 3 x 4 LHS RHS. Table x i Sum y j, j, i, n, i, n y 4 y 3 y 4 y 2 y 3 y 4 y 1 y 2 y 3 y 4 1 y 1 2 y 2 3 y 3 4 y 4 4 256 Expand y 1 4 256 1 32 y 1 3 y 2 3 32 y 1 2 y 2 2 1 8 y 1 y 2 3 y 2 4 16 3 64 y 1 3 y 3 9 9 16 y 1 y 2 2 y 3 3 8 y 2 3 y 3 27 128 y 1 2 y 3 2 27 32 y 1 y 2 y 3 2 27 27 64 y 1 y 3 3 27 32 y 2 y 3 3 4 81 y 3 256 1 16 y 1 3 y 4 3 8 y 1 2 y 2 y 4 32 y 1 2 y 2 y 3 32 y 2 2 y 3 2 3 4 y 1 y 2 2 y 4 1 2 y 2 3 y 4 9 16 y 1 2 y 3 y 4 5 y 1 y 2 y 3 y 4 4 5 4 y 2 2 y 3 y 4 11 16 y 1 y 3 2 y 4 11 8 y 2 y 3 2 y 4 11 16 y 3 3 y 4 3 8 y 1 2 y 4 2 1 2 y 1 y 2 y 4 2 1 2 y 2 2 y 4 2 1 4 y 1 y 3 y 4 2 1 2 y 2 y 3 y 4 2 3 8 y 3 2 y 4 2

Y-Positivity Y-Positivity Conjecture for Schur Functions If λ = µ and λ µ, then s λ (x) s λ (1) s µ(x) s µ (1) x i y i + y n is a polynomial in y with nonnegative coefficients.

Y-Positivity Y-Positivity Conjecture for Schur Functions If λ = µ and λ µ, then s λ (x) s λ (1) s µ(x) s µ (1) x i y i + y n is a polynomial in y with nonnegative coefficients. Proved for λ 9 and all n. (CG + Renggyi (Emily) Xu)

Symmetric Functions of Degree 3 Ultimate Problem: Classify all homogeneous symmetric function inequalities.

Symmetric Functions of Degree 3 More Modest Problem: Classify all homogeneous symmetric function inequalities of degree 3.

Symmetric Functions of Degree 3 This has long been recognized as an important question.

Symmetric Functions of Degree 3 Classifying all symmetric function inequalities of degree 3 We seek to characterize symmetric f (x) such that f (x) 0 for all x 0. Such f s will be called nonnegative. If f is homogeneous of degree 3 then f (x) = αm 3 (x) + βm 21 (x) + γm 111 (x), where the m s are monomial symmetric functions. Suppose that f (x) has n variables. Then the correspondence f (α, β, γ) parameterizes the set of nonnegative f s by a cone in R 3 with n extreme rays.

Symmetric Functions of Degree 3 Example: For example, if n = 3, there are three extreme rays, spanned by f 1 (x) = m 21 (x) 6m 111 (x) f 2 (x) = m 111 (x) f 3 (x) = m 3 (x) m 21 (x) + 3m 111 (x). If f is cubic, nonnegative, and symmetric in variables x = (x 1, x 2, x 3 ) then f may be expressed as a nonnegative linear combination of these three functions.

Symmetric Functions of Degree 3 Example: If n = 25, the cone looks like this: c 111 c 21 c 3

Symmetric Functions of Degree 3 Main Result: Theorem: If x = (x 1, x 2,..., x n ), and f (x) is a symmetric function of degree 3, then f (x) is nonnegative if and only if f (1 n k ) 0 for k = 1,..., n, where 1 n k = (1,..., 1, 0,..., 0), with k ones and (n k) zeros. Example: If x = (x 1, x 2, x 3 ) and f (x) = m 3 (x) m 21 (x) + 3m 111 (x), then f (1, 0, 0) = 1 f (1, 1, 0) = 2 2 = 0 f (1, 1, 1) = 3 6 + 3 = 0

Symmetric Functions of Degree 3 Main Result: Theorem: If x = (x 1, x 2,..., x n ), and f (x) is a symmetric function of degree 3, then f (x) is nonnegative if and only if f (1 n k ) 0 for k = 1,..., n, where 1 n k = (1,..., 1, 0,..., 0), with k ones and (n k) zeros. Example: If x = (x 1, x 2, x 3 ) and f (x) = m 3 (x) m 21 (x) + 3m 111 (x), then NOTES: f (1, 0, 0) = 1 f (1, 1, 0) = 2 2 = 0 f (1, 1, 1) = 3 6 + 3 = 0 The inequality f (x) 0 is known as Schur s Inequality (HLP). The statement analogous to the above theorem for degree d > 3 is false.

Symmetric Functions of Degree 3 Application: A positive function that is not y-positive Again take f (x) = m 3 (x) m 21 (x) + 3m 111 (x), but with n = 5 variables. Then f (1, 0, 0, 0, 0) = 1, f (1, 1, 0, 0, 0) = 0, f (1, 1, 1, 0, 0) = 0, f (1, 1, 1, 1, 0) = 4, f (1, 1, 1, 1, 1) = 15. Hence, by the Theorem, f (x) 0 for all x 0.

Symmetric Functions of Degree 3 Reference: Inequalities for Symmetric Functions of Degree 3, with Jeffrey Kroll, Jonathan Lima, Mark Skandera, and Rengyi Xu (to appear). Available on request, or at www.haverford.edu/math/cgreene.

Symmetric Functions of Degree 3 Acknowledgments Students who helped build two Mathematica packages that were used extensively: symfun.m: Julien Colvin, Ben Fineman, Renggyi (Emily) Xu posets.m: Eugenie Hunsicker, John Dollhopf, Sam Hsiao, Erica Greene