Small Littlewood-Richardson coefficients

Abstract

We develop structural insights into the Littlewood-Richardson graph, whose number of vertices equals the Littlewood-Richardson coefficient c(λ,μ,ν) for given partitions λ, μ, and ν. This graph was first introduced by Bürgisser and Ikenmeyer in arXiv:1204.2484, where its connectedness was proved. Our insights are useful for the design of algorithms for computing the Littlewood-Richardson coefficient: We design an algorithm for the exact computation of c(λ,μ,ν) with running time O(c(λ,μ,ν)^2 poly(n)), where λ, μ, and ν are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether c(λ,μ,ν) >= t whose running time is O(t^2 poly(n)). Even the existence of a polynomial-time algorithm for deciding whether c(λ,μ,ν) >= 2 is a nontrivial new result on its own. Our insights also lead to the proof of a conjecture by King, Tollu, and Toumazet posed in 2004, stating that c(λ,μ,ν) = 2 implies c(Mλ,Mμ,Mν) = M + 1 for all M. Here, the stretching of partitions is defined componentwise.