Small Littlewood-Richardson coefficients

Abstract

We develop structural insights into the Littlewood-Richardson graph, whose number of vertices equals the Littlewood-Richardson coefficient c({\lambda},{\mu},{\nu}) for given partitions {\lambda}, {\mu}, and {\nu}. This graph was first introduced by B\"urgisser 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({\lambda},{\mu},{\nu}) with running time O(c({\lambda},{\mu},{\nu})^2 poly(n)), where {\lambda}, {\mu}, and {\nu} are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether c({\lambda},{\mu},{\nu}) >= t whose running time is O(t^2 poly(n)). Even the existence of a polynomial-time algorithm for deciding whether c({\lambda},{\mu},{\nu}) >= 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({\lambda},{\mu},{\nu}) = 2 implies c(M{\lambda},M{\mu},M{\nu}) = M + 1 for all M. Here, the stretching of partitions is defined componentwise.