Are No More!

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 derive several more general statements using a recent result of Rhoades and Skandera. 1. Schur positivity conjectures The ring of symmetric functions has a linear basis of Schur functions sλ labelled by partitions λ = (λ1 ≥ λ2 ≥ · · · ≥ 0). A symmetric function is called Schur nonnegative if it is a linear combination with nonnegative coefficients of the Schur functions. In particular, skew Schur functions sλ/μ are Schur nonnegative. For two symmetric functions f and g, the notation f ≥s g means that f − g is Schur nonnegative. Recently, a lot of work has gone into studying whether certain expressions of the form sλsμ − sνsρ were Schur nonnegative. Let us mention several conjectures due to Okounkov, Fomin-Fulton-Li-Poon, and Lascoux-Leclerc-Thibon of this 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 that 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 , . . .). 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 formulate one of these conjectures, which was also studied recently by Bergeron and McNamara [BeMc]. For two partitions λ and μ, let λ ∪ μ = (ν1, ν2, ν3, . . . ) be the partition obtained by rearranging all parts of λ and μ in the weakly decreasing order. Let sort1(λ, μ) := (ν1, ν3, ν5, . . . ) and sort2(λ, μ) := (ν2, ν4, ν6, . . . ). Conjecture 2. Fomin-Fulton-Li-Poon [FFLP, Conjecture 2.8] For two partitions λ and μ, we have ssort1(λ,μ)ssort2(λ,μ) ≥s sλsμ. Date: February 18, 2005.