A BIJECTIVE PROOF OF r = 1 REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS

- Journal title : Honam Mathematical Journal
- Volume 32, Issue 2, 2010, pp.271-281
- Publisher : The Honam Mathematical Society
- DOI : 10.5831/HMJ.2010.32.2.271

Title & Authors

A BIJECTIVE PROOF OF r = 1 REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS

Moon, Dong-Ho;

Moon, Dong-Ho;

Abstract

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.

Keywords

Reduction formulae;Littlewood-Richardson coefficient;Schubert calculus;

Language

English

References

1.

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.

2.

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.

3.

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.

4.

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.

5.

Soojin Cho and Dongho Moon, Reduction formulae of Littlewood-Richardson coefficients, to appear in Advances in Applied Mathematics, 2010.

6.

William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997, With applications to representation theory and geometry.

7.

Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978.

8.

Christian Gutschwager, On multiplicity-free skew characters and the Schubert calculus, arXiv:math/0608145v2, to appear in Ann. Comb., 2006.

9.

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.

10.

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.

12.

Alexander A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (1998), no. 3, 419-445.

13.

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).

14.

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.