- Volume 32 Issue 2
Inspired by the reduction formulae between intersection numbers on Grassmannians obtained by Griffiths-Harris and the factorization theorem of Littlewood-Richardson coefficients by King, Tollu and Toumazet, eight reduction formulae has been discovered by the author and others. In this paper, we prove r = 1 reduction formula by constructing a bijective map between suitable sets of Littlewood-Richardson tableaux.
Reduction formulae;Littlewood-Richardson coefficient;Schubert calculus
- Soojin Cho, Eun-Kyoung Jung, and Dongho 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
- Soojin Cho, Eun-Kyoung Jung, and Dongho 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
- Soojin Cho, Eun-Kyoung Jung, and Dongho 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, DMTCS Proceedings, vol. AJ, 2008, pp. 483-494.
- Soojin Cho, Eun-Kyoung Jung, and Dongho Moon, An extension of reduction formula for littlewood-richardson coefficients, to appear in Journal of Korean Mathematical Society, 2010. https://doi.org/10.4134/JKMS.2010.47.6.1197
- Soojin Cho and Dongho Moon, Reduction formulae of Littlewood-Richardson coefficients, to appear in Advances in Applied Mathematics, 2010.
- William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997, With applications to representation theory and geometry.
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978.
- Christian Gutschwager, On multiplicity-free skew characters and the Schubert calculus, arXiv:math/0608145v2, to appear in Ann. Comb., 2006. https://doi.org/10.1007/s00026-010-0063-4
- Phil Hanlon and Sheila Sundaram, On a bijection between Littlewood-Richardson fillings of conjugate shape, J. Combin. Theory Ser. A 60 (1992), no. 1, 1-18. https://doi.org/10.1016/0097-3165(92)90034-R
- R. C. King, C. Tollu, and F. Toumazet, The hive model and the polynomial nature of stretched Littlewood-Richardson coefficients, Seminaire Lotharingien de Combinatoire 54A (2006), 1-19.
- R. C. King, C. Tollu, and F. Toumazet, J. Combin. Theory Ser. A 116 (2009), no. 2, 314-333. https://doi.org/10.1016/j.jcta.2008.06.005
- Alexander A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (1998), no. 3, 419-445. https://doi.org/10.1007/s000290050037
Allen Knutson, Terence Tao, and Christopher 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 (electronic). https://doi.org/10.1090/S0894-0347-03-00441-7
- Bruce E. Sagan, The symmetric group, second ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001, Representations, combinatorial algorithms, and symmetric functions.
- John R. Stembridge, Multiplicity-free products of Schur functions, Ann. Comb. 5 (2001), no. 2, 113-121. https://doi.org/10.1007/s00026-001-8008-6