L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy

Abstract

Let $\lambda$, $\mu$, ${\lambda }^{\prime }$, ${\mu }^{\prime }$ be partitions. The conjecture of Lam, Postnikov and Pylyavskyy states that, if $\lambda +\mu ={\lambda }^{\prime }+{\mu }^{\prime }$, and $\min ({\lambda }_i-{\lambda }_j,{\mu }_i-{\mu }_j)\le {\lambda }_i^{\prime }-{\lambda }_j^{\prime }\le \max ({\lambda }_i-{\lambda }_j,{\mu }_i-{\mu }_j)$ for all $1\le i<j\le n$, then $s_{{\lambda }^{\prime }}{s}_{{\mu }^{\prime }}-s_{\lambda }{s}_{\mu }$ is Schur nonnegative. We prove this conjecture. Our proof is based on two key ideas. First, we introduce a new combinatorial model for Littlewood-Richardson coefficients which we name ``skeps", which are similar to but distinct from Knutson and Tao's hives. Second, we use tools from Murota's theory of L-convexity to prove an L-log-concavity theorem for skeps.

v.2 corrects a minor error pointed out by Tuong Le. The main result is unchanged