Small Littlewood-Richardson coefficients

We develop structural insights into the Littlewood-Richardson graph, whose number of vertices equals the Littlewood-Richardson coefficient cνλ,μ for given partitions λ, μ and ν. This graph was first introduced in [BI12], 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 cνλ,μ with running time O ( (cνλ,μ) 2 · poly(n) ) , where λ, μ, and ν are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether cνλ,μ ≥ t whose running time is O ( t · poly(n) ) . Even the existence of a polynomial-time algorithm for deciding whether cνλ,μ ≥ 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 [KTT04], stating that cνλ,μ = 2 implies c Mν Mλ,Mμ = M + 1 for all M ∈ N. Here, the stretching of partitions is defined componentwise. 2010 MSC: 05E10, 22E46, 90C27