Schur positivity and Schur log-concavity

Abstract

We prove Okounkov's conjecture, a conjecture of Fomin-Fulton-Li-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.

Necessary conditions for Schur-positivity

“…• rects k,l (A) ≤ rects k,l (B) for all k, l. for partitions α, β has been studied, for example, in [1,4,6]. It is natural to ask what Theorem 3.8 tells us about expressions of the form (4.1).…”

“…For example, skew Schur functions s λ/µ and the product s σ s τ of two Schur functions are famous examples of Schur-positive functions: they can be written as linear combinations of Schur functions with all coefficients positive. Taking this a step further, several recent papers such as [1,4,5,6,7,10] have asked when expressions of the form s λ/µ − s σ/τ or s λ s µ − s σ s τ are Schur-positive. These papers have been concerned with giving conditions on λ, µ, σ and τ that result in Schur-positive expressions.…”

Bitableaux and Zero Sets of Dual Canonical Basis Elements

“…(See, e.g., [7], [14], [16], [31], [50], [51].) Furthermore, just as the structure constants describing multiplication of Kazhdan-Lusztig basis elements in C[S n ] belong to N, so do those describing multiplication…”

“…Since then, nonnegativity properties of Kazhdan-Lusztig immanants studied in [50] have played an important role in establishing inequalities satisfied by Littlewood-Richardson coefficients [31] and in creating a representation theoretic model for the combinatorial action of jeu-detaquin promotion [48], thus proving conjectures of Fomin-Fulton-Li-Poon [20], ReinerStanton-White [47] and others.…”

Log-convexity properties of Schur functions and generalized hypergeometric functions of matrix argument

“…In some recent articles Lam, et al [12,13] have established some interesting positivity properties of the Schur functions. A consequence of the work of Lam, et al [12,13] is the following result.…”