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

Abstract

Let $\lambda$, $\mu$, $\lambda'$, $\mu'$ be partitions. The conjecture of Lam, Postnikov and Pylyavskyy states that, if $\lambda+\mu = \lambda' + \mu'$, and $\min(\lambda_i-\lambda_j, \mu_i-\mu_j) \leq \lambda'_i - \lambda'_j \leq \max(\lambda_i-\lambda_j, \mu_i-\mu_j)$ for all $1 \leq i<j \leq n$, then $s_{\lambda'} s_{\mu'} - s_{\lambda} s_{\mu}$ 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.