Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Recent progress in algebraic combinatorics


Author: Richard P. Stanley
Journal: Bull. Amer. Math. Soc. 40 (2003), 55-68
MSC (2000): Primary 05E99; Secondary 05E05, 14C05, 15A18, 60C05
DOI: https://doi.org/10.1090/S0273-0979-02-00966-7
Published electronically: October 11, 2002
MathSciNet review: 1943133
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We survey three recent breakthroughs in algebraic combinatorics. The first is the proof by Knutson and Tao, and later Derksen and Weyman, of the saturation conjecture for Littlewood-Richardson coefficients. The second is the proof of the $n!$ and $(n+1)^{n-1}$ conjectures by Haiman. The final breakthrough is the determination by Baik, Deift, and Johansson of the limiting behavior of the length of the longest increasing subsequence of a random permutation.


References [Enhancements On Off] (What's this?)

  • 1. J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), 1119-1178, math.CO/9810105.[*] MR 2000e:05006
  • 2. A. Berenstein and A. Zelevinsky, Triple multiplicities for $sl(r+1)$ and the spectrum of the exterior algebra of the adjoint representation, J. Algebraic Combinatorics 1 (1992), 7-22. MR 93h:17012
  • 3. F. Bergeron, A. Garsia, and G. Tesler, Multiple left regular representations generated by alternants, J. Combinatorial Theory (A) 91 (2000), 49-83. MR 2002c:05158
  • 4. A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481-515, math.CO/9905032. MR 2001g:05103
  • 5. A. Buch, The saturation conjecture (after A. Knutson and T. Tao), Enseign. Math. 46 (2000), 43-60, math.CO/9810180. MR 2001g:05105
  • 6. P. Deift, Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), 631-640. MR 2001g:05012
  • 7. H. Derksen and J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, J. Amer. Math. Soc. 13 (2000), 467-479. MR 2001g:16031
  • 8. W. Fulton, Young Tableaux, London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge, 1997. MR 99f:05119
  • 9. W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. 37 (2000), 209-249, math.AG/9908012. MR 2001g:15023
  • 10. A. M. Garsia and J. Haglund, A proof of the $q,t$-Catalan positivity conjecture, Discrete Math., to appear, www.math.upenn.edu/$\sim$jhaglund.
  • 11. A. M. Garsia and M. Haiman, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191-244. MR 97k:05208
  • 12. A. M. Garsia and M. Haiman, A graded representation model for Macdonald's polynomials, Proc. Nat. Acad. Sci. U.S.A. 90 (1993), 3607-3610. MR 94b:05206
  • 13. A. M. Garsia and M. Haiman, Some natural bigraded $S_n$-modules and $q,t$-Kostka coefficients, Electron. J. Combin. 3 (1996), RP24. MR 97k:05205
  • 14. I. Gessel, Symmetric functions and P-recursiveness, J. Combinatorial Theory (A) 53 (1990), 257-285. MR 91c:05190
  • 15. C. Greene, An extension of Schensted's theorem, Advances in Math. 14 (1974), 254-265. MR 50:6874
  • 16. J. Haglund, Conjectured statistics for the $q,t$-Catalan numbers, Advances in Math., to appear, www.math.upenn.edu/$\sim$jhaglund.
  • 17. M. Haiman, Macdonald polynomials and geometry, in New perspectives in algebraic combinatorics (Berkeley, CA, 1996-97) (L. J. Billera, et al., eds.), MSRI Publ. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 207-254. MR 2001k:05203
  • 18. M. Haiman, Hilbert schemes, polygraphs, and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941-1006, www/math.berkeley.edu/$\sim$mhaiman. MR 2002c:14008
  • 19. M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, preliminary draft, www/math.berkeley.edu/$\sim$mhaiman; abbreviated version in Physics and Combinatorics (A. N. Kirillov and N. Liskova, eds.), World Scientific, London, 2001, pp. 1-21.
  • 20. P. Hall, The algebra of partitions, in Proc. 4th Canadian Math. Congress (Banff), 1959, pp. 147-159.
  • 21. J. M. Hammersley, A few seedlings of research, in Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California Press, Berkeley/Los Angeles, 1972, pp. 345-394. MR 53:9457
  • 22. G. J. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Invent. Math. 67 (1982), 333-356. MR 84d:22019
  • 23. K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. 153 (2001), 259-296, math.CO/9906120. MR 2002g:05188
  • 24. A. A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. 4 (1998), 419-445. MR 2000b:14054
  • 25. D. E. Knuth, The Art of Computer Programming, vol. 3, Sorting and Searching, Addison-Wesley, Reading, Massachusetts, 1973; second edition, 1998. MR 56:4281
  • 26. A. Knutson and T. Tao, The honeycomb model of GL $_n(\mathbb{C} )$ tensor products I: Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), 1055-1090, math.RT/9807160. MR 2000c:20066
  • 27. A. Knutson and T. Tao, Honeycombs and sums of Hermitian matrices, Notices Amer. Math. Soc. 48 (2001), 175-186, math.RT/0009048. MR 2002g:15020
  • 28. B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206-222. MR 98e:05108 [sic]
  • 29. I. G. Macdonald, A new class of symmetric functions, Actes 20 $^\mathrm{e}$ Séminaire Lotharingien, Publ. I.R.M.A., Strasbourg, 1992, pp. 5-39.
  • 30. I. G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford University Press, Oxford, 1995. MR 96h:05207
  • 31. F. M. Maley, The Hall polynomial revisited, J. Algebra 184 (1996), 363-371. MR 97j:20054
  • 32. A. Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices 2000, 1043-1095, math.CO/9903176. MR 2002c:15045
  • 33. D. Rotem, On a correspondence between binary trees and a certain type of permutation, Inf. Proc. Letters 4 (1975/76), 58-61. MR 52:9675
  • 34. C. E. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961), 179-191. MR 22:12047
  • 35. L. Smith, Polynomial Invariants of Finite Groups, A K Peters, Wellesley, Massachusetts, 1995. MR 96f:13008
  • 36. R. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (new series) 1 (1979), 475-511. MR 81a:20015
  • 37. R. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999. MR 2000k:05026
  • 38. G. Tesler, Semi-primary lattices and tableaux algorithms, Ph.D. thesis, M.I.T., 1995.
  • 39. C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151-174, hep-th/9211141. MR 95e:82003
  • 40. A. M. Vershik and S. V. Kerov, Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Dokl. Akad. Nauk SSSR 233 (1977), 1024-1027. English translation in Soviet Math. Dokl. 18 (1977), 527-531. MR 58:562
  • 41. A. Zelevinsky, Littlewood-Richardson semigroups, in New perspectives in algebraic combinatorics (Berkeley, CA, 1996-97) (L. J. Billera, et al., eds.), MSRI Publ. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 337-345, math.CO/9704228. MR 2000j:05126

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 05E99, 05E05, 14C05, 15A18, 60C05

Retrieve articles in all journals with MSC (2000): 05E99, 05E05, 14C05, 15A18, 60C05


Additional Information

Richard P. Stanley
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: rstan@math.mit.edu

DOI: https://doi.org/10.1090/S0273-0979-02-00966-7
Received by editor(s): October 23, 2000
Received by editor(s) in revised form: January 4, 2002
Published electronically: October 11, 2002
Additional Notes: Partially supported by NSF grant #DMS-9988459
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society