A BIJECTIVE PROOF OF THE SECOND REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 45, Issue 3, 2008, pp.485-494
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2008.45.3.485

Title & Authors

A BIJECTIVE PROOF OF THE SECOND REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS

Cho, Soo-Jin; Jung, Eun-Kyoung; Moon, Dong-Ho;

Cho, Soo-Jin; Jung, Eun-Kyoung; Moon, Dong-Ho;

Abstract

There are two well known reduction formulae for structural constants of the cohomology ring of Grassmannians, i.e., Littlewood-Richardson coefficients. Two reduction formulae are a conjugate pair in the sense that indexing partitions of one formula are conjugate to those of the other formula. A nice bijective proof of the first reduction formula is given in the authors' previous paper while a (combinatorial) proof for the second reduction formula in the paper depends on the identity between Littlewood-Richardson coefficients of conjugate shape. In this article, a direct bijective proof for the second reduction formula for Littlewood-Richardson coefficients is given. Our proof is independent of any previously known results (or bijections) on tableaux theory and supplements the arguments on bijective proofs of reduction formulae in the authors' previous paper.

Keywords

reduction formulae;Littlewood-Richardson coefficient;Schubert calculus;

Language

English

Cited by

1.

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

1.

2.

3.

References

1.

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

2.

S. Cho, E.-K. Jung, and D. Moon, Some cases of King's conjecture on factorization of Littlewood-Richardson polynomials, preprint, 2007

3.

I. Coskun, A Littlewood-Richardson rule for two-step flag varieties, Preprint, http: //www-math.mit.edu/-coskun/reviki51.pdf

4.

I. Coskun and R. Vakil, Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus, Preprint, http://www-math.mit.edu/-coskun/seattleoct17.pdf

5.

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

6.

P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience John Wiley & Sons, New York, 1978

7.

P. Hanlon and S. Sundaram, On a bijection between Littlewood-Richardson fillings of conjugate shape, J. Combin. Theory Ser. A 60 (1992), no. 1, 1-18

8.

J. Harris, Algebraic Geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995

9.

R. C. King, C. Tollu, and F. Toumazet, Stretched Littlewood-Richardson and Kostka coefficients, Symmetry in physics, CRM Proc. Lecture Notes, vol. 34, Amer. Math. Soc., Providence, RI, 2004, pp. 99-112

10.

R. C. King, C. Tollu, and F. Toumazet, The hive model and the polynomial nature of stretched Littlewood-Richardson coefficients, Sem. Lothar. Combin. 54A (2006), 1-19

11.

R. C. King, C. Tollu, and F. Toumazet, Factorization of Littlewood-Richardson coefficients, preprint, 2007

12.

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

13.

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

14.

D. E. Littlewood and A. R. Richardson, Group characters and algebra, Phi. Trans. A (1934), 99-141

15.

R. P. Stanley, Enumerative Combinatorics. Vol. 2, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999