2 00 5 Schur Positivity and Schur Log - Concavity

We prove Okounkov’s conjecture, a conjecture of Fomin-FultonLi-Poon, and a special case of Lascoux-Leclerc-Thibon’s conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. An alternative proof of this result is provided. We also give an intriguing log-concavity property of Schur functions. 1. Schur positivity conjectures The ring of symmetric functions has a linear basis of Schur functions sλ labelled by partitions λ = (λ1 ≥ λ2 ≥ · · · ≥ 0), see [Mac]. These functions appear in representation theory as characters of irreducible representations of GLn and in geometry as representatives of Schubert classes for Grassmannians. A symmetric function is called Schur nonnegative if it is a linear combination with nonnegative coefficients of the Schur functions, or, equivalently, if it is the character of a certain representation of GLn. In particular, skew Schur functions sλ/μ are Schur nonnegative. Recently, a lot of work has gone into studying whether certain expressions of the form sλsμ − sνsρ were Schur nonnegative. Schur positivity of an expression of this form is equivalent to some inequalities between Littlewood-Richardson coefficients. In a sense, characterizing such inequalities is a “higher analogue” of the Klyachko problem on nonzero Littlewood-Richardson coefficients. Let us mention several Schur positivity conjectures due to Okounkov, Fomin-Fulton-Li-Poon, and Lascoux-Leclerc-Thibon of the above form. Okounkov [Oko] studied branching rules for classical Lie groups and proved that the multiplicities were “monomial log-concave” in some sense. An essential combinatorial ingredient in his construction was the theorem about monomial nonnegativity of some symmetric functions. He conjectured that these functions are Schur nonnegative, as well. For a partition λ with all even parts, let λ2 denote the partition (1 2 , λ2 2 , . . .). For two symmetric functions f and g, the notation f ≥s g means that f − g is Schur nonnegative. Conjecture 1. Okounkov [Oko] For two skew shapes λ/μ and ν/ρ such that λ+ν and μ+ ρ both have all even parts, we have (s (λ+ν) 2 / (μ+ρ) 2 ) ≥s sλ/μ sν/ρ. Fomin, Fulton, Li, and Poon [FFLP] studied the eigenvalues and singular values of sums of Hermitian and of complex matrices. Their study led to two combinatorial conjectures concerning differences of products of Schur functions. Let us Date: February 18, 2005, updated September 26, 2005.