Inequalities for Symmetric Polynomials Curtis Greene October 24, - PowerPoint PPT Presentation

Inequalities for Symmetric Polynomials Inequalities for Symmetric Polynomials Curtis Greene October 24, 2009

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

Inequalities for Symmetric Polynomials Inequalities for Averages and Means Classical examples (e.g., Hardy-Littlewood-Polya) THE AGM INEQUALITY: x 1 + x 2 + · · · + x n ≥ ( x 1 x 2 · · · x n ) 1 / n ∀ x ≥ 0. n NEWTON’S INEQUALITIES: e k ( x ) e k ( x ) e k ( 1 ) ≥ e k − 1 ( x ) e k +1 ( x ) ∀ x ≥ 0 e k ( 1 ) e k − 1 ( 1 ) e k +1 ( 1 )

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

Inequalities for Symmetric Polynomials 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 ) SCHL¨ OMILCH’S (POWER SUM) INEQUALITIES: � p j ( x ) � 1 / j ≤ � p k ( x ) � 1 / k if j ≤ k , x ≥ 0 n n

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

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

Inequalities for Symmetric Polynomials 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 Symmetric Polynomials 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 . CONJECTURE: the last two implications are ⇐ ⇒ . Reference: Cuttler,Greene, Skandera

Inequalities for Symmetric Polynomials 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 .)

Inequalities for Symmetric Polynomials 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 . λ µ NORMALIZED MAJORIZATION: λ ⊑ µ iff | λ | � | 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 .

Inequalities for Symmetric Polynomials 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 . λ µ NORMALIZED MAJORIZATION: λ ⊑ µ iff | λ | � | 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.

Inequalities for Symmetric Polynomials 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.

Inequalities for Symmetric Polynomials 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 ⇐ ⇒ .

Inequalities for Symmetric Polynomials 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 ?

Inequalities for Symmetric Polynomials 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 ? We have no idea.

Inequalities for Symmetric Polynomials 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 ? We have no idea. What about inequalities for monomial means M λ ?

Inequalities for Symmetric Polynomials 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 ? We have no idea. What about inequalities for monomial means M λ ? We know a lot.

Inequalities for Symmetric Polynomials 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).

Inequalities for Symmetric Polynomials 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.

Inequalities for Symmetric Polynomials Master Theorem: Double Majorization The double (normalized) majorization order ⊤ ⊒ µ ⊤ DEFINITION: λ � µ iff λ ⊑ µ and λ , ⊤ µ | λ | � µ ⊤ λ | µ | and λ EQUIVALENTLY: λ � µ iff | λ | � | µ | . DEFINITION: DP ∗ = ( P ∗ , � )

Inequalities for Symmetric Polynomials Master Theorem: Double Majorization The double (normalized) majorization order ⊤ ⊒ µ ⊤ DEFINITION: λ � µ iff λ ⊑ µ and λ , ⊤ µ | λ | � µ ⊤ λ | µ | and λ EQUIVALENTLY: λ � µ iff | λ | � | µ | . DEFINITION: DP ∗ = ( P ∗ , � ) NOTES: ⊤ are not equivalent. ⊤ ⊒ µ ◮ The conditions λ ⊑ µ and λ 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.

Inequalities for Symmetric Polynomials 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 .)