Journal of Algebraic Combinatorics

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

Small Littlewood–Richardson coefficients

  • Christian Ikenmeyer


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.


Littlewood–Richardson coefficient Hive model Efficient algorithms Flows in networks 

Mathematics Subject Classification

05E10 22E46 90C27 



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.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall Inc, Upper Saddle River (1993)MATHGoogle Scholar
  2. 2.
    Bürgisser, P., Ikenmeyer, C.: Deciding positivity of Littlewood–Richardson coefficients. SIAM J. Discrete Math. 27(4), 1639–1681 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Buch, A.S.: The saturation conjecture (after A. Knutson and T. Tao) with an appendix by William Fulton. Enseign. Math. 2(46), 43–60 (2000)MathSciNetMATHGoogle Scholar
  4. 4.
    Berenstein, A.D., Zelevinsky, A.V.: Triple multiplicities for sl(r + 1) and the spectrum of the exterior algebra of the adjoint representation. J. Algebr. Comb. 1(1), 7–22 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    De Loera, J.A., McAllister, T.B.: On the computation of Clebsch–Gordan coefficients and the dilation effect. Exp. Math. 15(1), 7–19 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fukuda, K., Matsui, T.: Finding all the perfect matchings in bipartite graphs. Appl. Math. Lett. 7(1), 15–18 (1994)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fulton, W.: Young Tableaux, volume 35 of London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)Google Scholar
  8. 8.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  9. 9.
    Knutson, A., Tao, T.: The honeycomb model of \({\rm GL}_n({\rm C})\) tensor products. I. Proof of the saturation conjecture. J. Am. Math. Soc. 12(4), 1055–1090 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Knutson, Allen, Tao, Terence: Honeycombs and sums of Hermitian matrices. Notices Am. Math. Soc. 48(2), 175–186 (2001)MathSciNetMATHGoogle Scholar
  11. 11.
    King, R.C., Tollu, C., Toumazet, F.: Stretched Littlewood-Richardson and Kostka coefficients. In: Symmetry in physics, volume 34 of CRM Proc. Lecture Notes, pp. 99–112. American Mathematical Society, Providence, RI (2004)Google Scholar
  12. 12.
    Ketan, D.: Mulmuley and Milind Sohoni. Geometric complexity theory III: On deciding positivity of Littlewood–Richardson coefficients. cs.ArXive preprint cs.CC/0501076 (2005)Google Scholar
  13. 13.
    Narayanan, Hariharan: On the complexity of computing Kostka numbers and Littlewood–Richardson coefficients. J. Algebr. Combin. 24(3), 347–354 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Tardos, Éva: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34(2), 250–256 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Texas A&M UniversityCollege StationUSA

Personalised recommendations