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Journal of Algebraic Combinatorics

, Volume 44, Issue 1, pp 1–29 | Cite as

Small Littlewood–Richardson coefficients

  • Christian Ikenmeyer
Article
  • 179 Downloads

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 in Bürgisser 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 \(c_{\lambda ,\mu }^{\nu }\) with running time \(\mathcal {O}\big ((c_{\lambda ,\mu }^{\nu })^2 \cdot {\textsf {poly}}(n)\big )\), where \(\lambda \), \(\mu \), and \(\nu \) are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge t\) whose running time is \(\mathcal {O}\big (t^2 \cdot {\textsf {poly}}(n)\big )\). Even the existence of a polynomial-time algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge 2\) 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 \(c_{\lambda ,\mu }^{\nu }=2\) implies \(c_{M\lambda ,M\mu }^{M\nu } = M+1\) for all \(M \in \mathbb {N}\). Here, the stretching of partitions is defined componentwise.

Keywords

Littlewood–Richardson coefficient Hive model Efficient algorithms Flows in networks 

Mathematics Subject Classification

05E10 22E46 90C27 

Notes

Acknowledgments

This research was conducted at the University of Paderborn. I benefitted tremendously from the long, intense, and invaluable discussions with my Ph.D. advisor Peter Bürgisser. I thank the Deutsche Forschungsgemeinschaft for their financial support (DFG-Grants BU 1371/3-1 and BU 1371/3-2). Furthermore, I thank an anonymous reviewer for his/her dedication to read this paper in detail and for giving valuable suggestions concerning its exposition.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Texas A&M UniversityCollege StationUSA

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