# AN EXTENSION OF REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS

• Cho, Soo-Jin ;
• Jung, Eun-Kyoung ;
• Moon, Dong-Ho
• Published : 2010.11.01
• 46 5

#### Abstract

There is a well-known classical reduction formula by Griffiths and Harris for Littlewood-Richardson coefficients, which reduces one part from each partition. In this article, we consider an extension of the reduction formula reducing two parts from each partition. This extension is a special case of the factorization theorem of Littlewood-Richardson coefficients by King, Tollu, and Toumazet (the KTT theorem). This case of the KTT factorization theorem is of particular interest, because, in this case, the KTT theorem is simply a reduction formula reducing two parts from each partition. A bijective proof using tableaux of this reduction formula is given in this paper while the KTT theorem is proved using hives.

#### Keywords

Littlewood-Richardson coefficients;Reduction formulae

#### References

1. A. S. Buch, The saturation conjecture (after A. Knutson and T. Tao), With an appendix by William Fulton, Enseign. Math. (2) 46 (2000), no. 1-2, 43-60.
2. A. S. Buch, A. Kresch, and H. Tamvakis, Littlewood-Richardson rules for Grassmannians, Adv. Math. 185 (2004), no. 1, 80-90. https://doi.org/10.1016/S0001-8708(03)00165-8
3. S. Cho, E.-K. Jung, and D. Moon, A combinatorial proof of the reduction formula for Littlewood-Richardson coefficients, J. Combin. Theory Ser. A 114 (2007), no. 7, 1199-1219. https://doi.org/10.1016/j.jcta.2007.01.003
4. S. Cho, E.-K. Jung, and D. Moon, Reduction formulae from the factorization Theorem of Littlewood-Richardson polynomials by King, Tollu, and Toumazet, 20th International Conference on Formal Power Series and Algebraic Combinatorics, AJ, DMTCS Proceedings, 483-494, 2008.
5. S. Cho, E.-K. Jung, and D. Moon, A bijective proof of the second reduction formula for Littlewood-Richardson coefficients, Bull. Korean Math. Soc. 45 (2008), no. 3, 485-494. https://doi.org/10.4134/BKMS.2008.45.3.485
6. S. Cho, E.-K. Jung, and D. Moon, Reduction formulae of Littlewood-Richardson coefficients, submitted for publication.
7. H. Derksen and J. Weyman, On the Littlewood-Richardson polynomials, J. Algebra 255 (2002), no. 2, 247-257. https://doi.org/10.1016/S0021-8693(02)00125-4
8. W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209-249. https://doi.org/10.1090/S0273-0979-00-00865-X
9. W. Fulton, Young Tableaux, With applications to representation theory and geometry. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997.
10. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978.
11. J. Harris, Algebraic Geometry, Graduate Texts in Mathematics, 133, Springer-Verlag, New York, 1995.
12. R. C. King, The hive model and the polynomial nature of stretched Littlewood-Richardson coefficients, Proceedings of the 17th Conference on Formal Power Series and Algebraic Combinatorics, held at the University of Messina, Taormina, Italy, June 20-25, 2005.
13. R. C. King, C. Tollu, and F. Toumazet, Stretched Littlewood-Richardson and Kostka coefficients, Symmetry in physics, 99-112, CRM Proc. Lecture Notes, 34, Amer. Math. Soc., Providence, RI, 2004.
14. R. C. King, C. Tollu, and F. Toumazet, The hive model and the polynomial nature of stretched Littlewood-Richardson Coefficients, S´eminaire Lotharingien de Combinatoire 54A (2006), 1-19.
15. R. C. King, C. Tollu, and F. Toumazet, Factorisation of Littlewood-Richardson coefficients, J. Combin. Theory Ser. A 116 (2009), no. 2, 314-333. https://doi.org/10.1016/j.jcta.2008.06.005
16. A. Knutson and T. Tao, The honeycomb model of $GL_n(C)$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055-1090. https://doi.org/10.1090/S0894-0347-99-00299-4
17. A. Knutson, T. Tao, and C. Woodward, The honeycomb model of $GL_n(C)$ tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc. 17 (2004), no. 1, 19-48 https://doi.org/10.1090/S0894-0347-03-00441-7
18. Marc A. A. van Leeuwen, The Littlewood-Richardson rule, and related combinatorics, Interaction of combinatorics and representation theory, 95-145, MSJ Mem., 11, Math. Soc. Japan, Tokyo, 2001.
19. D. E. Littlewood and A. R. Richardson, Group characters and algebra, Phi. Trans. A 233 (1934), 99-141. https://doi.org/10.1098/rsta.1934.0015
20. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1979.
21. E. Rassart, A polynomiality property for Littlewood-Richardson coefficients, J. Combin. Theory Ser. A 107 (2004), no. 2, 161-179. https://doi.org/10.1016/j.jcta.2004.04.003
22. J. B. Remmel and M. Shimozono, A simple proof of the Littlewood-Richardson rule and applications, Selected papers in honor of Adriano Garsia (Taormina, 1994). Discrete Math. 193 (1998), no. 1-3, 257-266. https://doi.org/10.1016/S0012-365X(98)00145-9
23. G. de B. Robinson, On the representations of the symmetric group, Amer. J. Math. 60 (1938), no. 3, 745-760. https://doi.org/10.2307/2371609
24. I. Schur, Uber die rationalen Darstellungen der allgemeinen linearen Gruppe, Preuss. Akad. Wiss. Sitz. (1927), 58-75
25. M.-P. Schutzenberger, La correspondance de Robinson, Combinatoire et representation du groupe symetrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), pp. 59-113.
26. Igor R. Shafarevich, Basic Algebraic Geometry. 2, Schemes and complex manifolds. Second edition. Translated from the 1988 Russian edition by Miles Reid. Springer-Verlag, Berlin, 1994.
27. J. R. Stembridge, A concise proof of the Littlewood-Richardson rule, Electron. J. Combin. 9 (2002), no. 1, Note 5, 4 pp.
28. G. P. Thomas, On Schensted’s construction and the multiplication of Schur functions, Adv. in Math. 30 (1978), no. 1, 8-32. https://doi.org/10.1016/0001-8708(78)90129-9
29. I. Schur, Uber die rationalen Darstellungen der allgemeinen linearen Gruppe, reprinted in Gesamelte Abhandlungen vol. 3, 68-85.
30. M.-P. Schutzenberger, La correspondance de Robinson, Combinatoire et representation du groupe symetrique, pp. 59-113. Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977. https://doi.org/10.1007/BFb0090012

#### Cited by

1. Reduction formulae of Littlewood–Richardson coefficients vol.46, pp.1-4, 2011, https://doi.org/10.1016/j.aam.2009.12.005
2. A BIJECTIVE PROOF OF r = 1 REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS vol.32, pp.2, 2010, https://doi.org/10.5831/HMJ.2010.32.2.271