DOI QR코드

DOI QR Code

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

  • Published : 2008.08.31

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

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 https://doi.org/10.1016/j.jcta.2007.01.003
  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 and R. Vakil, Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus, Preprint, http://www-math.mit.edu/-coskun/seattleoct17.pdf
  4. W. Fulton, Young Tableaux, With applications to representation theory and geometry. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997
  5. 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 https://doi.org/10.1016/0097-3165(92)90034-R
  6. J. Harris, Algebraic Geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995
  7. 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
  8. R. C. King, C. Tollu, and F. Toumazet, Factorization of Littlewood-Richardson coefficients, preprint, 2007
  9. 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
  10. 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
  11. D. E. Littlewood and A. R. Richardson, Group characters and algebra, Phi. Trans. A (1934), 99-141
  12. R. Vakil, A geometric Littlewood-Richardson rule, Appendix A written with A. Knutson. Ann. of Math. (2) 164 (2006), no. 2, 371-421 https://doi.org/10.4007/annals.2006.164.371
  13. I. Coskun, A Littlewood-Richardson rule for two-step flag varieties, Preprint, http: //www-math.mit.edu/-coskun/reviki51.pdf
  14. P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience John Wiley & Sons, New York, 1978
  15. 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
  16. 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

Cited by

  1. AN EXTENSION OF REDUCTION FORMULA FOR LITTLEWOOD-RICHARDSON COEFFICIENTS vol.47, pp.6, 2010, https://doi.org/10.4134/JKMS.2010.47.6.1197
  2. Reduction formulae of Littlewood–Richardson coefficients vol.46, pp.1-4, 2011, https://doi.org/10.1016/j.aam.2009.12.005
  3. 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
  4. A Hive-Model Proof of the Second Reduction Formula of Littlewood-Richardson Coefficients vol.15, pp.2, 2011, https://doi.org/10.1007/s00026-011-0091-8