L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy (Combinatorics Seminar)

Let lambda, mu, lambda', mu' be partitions. The conjecture of Lam, Postnikov and Pylyavskyy states that, if (lambda, mu, lambda', mu') obey certain natural inequalities, 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.