Small Littlewood–Richardson coefficients

Abstract

We develop structural insights into the Littlewood---Richardson graph, whose number of vertices equals the Littlewood---Richardson coefficient $$c_{\lambda ,\mu }^{\nu }$$c?,μ? for given partitions $$\lambda $$?, $$\mu $$μ, and $$\nu $$?. This graph was first introduced in Burgisser and Ikenmeyer (SIAM J Discrete Math 27(4):1639---1681, 2013), 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 $$c_{\lambda ,\mu }^{\nu }$$c?,μ? with running time $$\mathcal {O}\big ((c_{\lambda ,\mu }^{\nu })^2 \cdot {\textsf {poly}}(n)\big )$$O((c?,μ?)2·poly(n)), where $$\lambda $$?, $$\mu $$μ, and $$\nu $$? are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether $$c_{\lambda ,\mu }^{\nu } \ge t$$c?,μ??t whose running time is $$\mathcal {O}\big (t^2 \cdot {\textsf {poly}}(n)\big )$$O(t2·poly(n)). Even the existence of a polynomial-time algorithm for deciding whether $$c_{\lambda ,\mu }^{\nu } \ge 2$$c?,μ??2 is a nontrivial new result on its own. Our insights also lead to the proof of a conjecture by King et al. (Symmetry in physics. American Mathematical Society, Providence, 2004), stating that $$c_{\lambda ,\mu }^{\nu }=2$$c?,μ?=2 implies $$c_{M\lambda ,M\mu }^{M\nu } = M+1$$cM?,MμM?=M+1 for all $$M \in \mathbb {N}$$M?N. Here, the stretching of partitions is defined componentwise.