# Small Littlewood–Richardson coefficients

- 172 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.

## References

- 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.Bürgisser, P., Ikenmeyer, C.: Deciding positivity of Littlewood–Richardson coefficients. SIAM J. Discrete Math.
**27**(4), 1639–1681 (2013)MathSciNetCrossRefMATHGoogle Scholar - 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.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.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.Fukuda, K., Matsui, T.: Finding all the perfect matchings in bipartite graphs. Appl. Math. Lett.
**7**(1), 15–18 (1994)MathSciNetCrossRefMATHGoogle Scholar - 7.Fulton, W.: Young Tableaux, volume 35 of London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)Google Scholar
- 8.Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
- 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.Knutson, Allen, Tao, Terence: Honeycombs and sums of Hermitian matrices. Notices Am. Math. Soc.
**48**(2), 175–186 (2001)MathSciNetMATHGoogle Scholar - 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.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.Narayanan, Hariharan: On the complexity of computing Kostka numbers and Littlewood–Richardson coefficients. J. Algebr. Combin.
**24**(3), 347–354 (2006)MathSciNetCrossRefMATHGoogle Scholar - 14.Tardos, Éva: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res.
**34**(2), 250–256 (1986)MathSciNetCrossRefMATHGoogle Scholar