Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

A simple proof of the Zeilberger-Bressoud $ q$-Dyson theorem


Authors: Gyula Károlyi and Zoltán Lóránt Nagy
Journal: Proc. Amer. Math. Soc. 142 (2014), 3007-3011
MSC (2010): Primary 05A19, 05A30, 33D05, 33D60
DOI: https://doi.org/10.1090/S0002-9939-2014-12041-7
Published electronically: May 28, 2014
MathSciNet review: 3223356
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: As an application of the Combinatorial Nullstellensatz, we give a short polynomial proof of the $ q$-analogue of Dyson's conjecture formulated by Andrews and first proved by Zeilberger and Bressoud.


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

  • [1] Noga Alon, Combinatorial Nullstellensatz, Recent trends in combinatorics (Mátraháza, 1995), Combin. Probab. Comput. 8 (1999), no. 1-2, 7-29. MR 1684621 (2000b:05001), https://doi.org/10.1017/S0963548398003411
  • [2] George E. Andrews, Problems and prospects for basic hypergeometric functions, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), Academic Press, New York, 1975, pp. 191-224. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. MR 0399528 (53 #3372)
  • [3] T. H. Baker and P. J. Forrester, Generalizations of the $ q$-Morris constant term identity, J. Combin. Theory Ser. A 81 (1998), no. 1, 69-87. MR 1492869 (99e:05013), https://doi.org/10.1006/jcta.1997.2819
  • [4] D. M. Bressoud and I. P. Goulden, Constant term identities extending the $ q$-Dyson theorem, Trans. Amer. Math. Soc. 291 (1985), no. 1, 203-228. MR 797055 (86k:05011), https://doi.org/10.2307/1999904
  • [5] Ivan Cherednik, Double affine Hecke algebras and Macdonald's conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191-216. MR 1314036 (96m:33010), https://doi.org/10.2307/2118632
  • [6] Freeman J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962), 140-156. MR 0143556 (26 #1111)
  • [7] P. J. Forrester, Normalization of the wavefunction for the Calogero-Sutherland model with internal degrees of freedom, Internat. J. Modern Phys. B 9 (1995), no. 10, 1243-1261. MR 1338022 (96e:82005), https://doi.org/10.1142/S0217979295000537
  • [8] Ira M. Gessel and Guoce Xin, A short proof of the Zeilberger-Bressoud $ q$-Dyson theorem, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2179-2187 (electronic). MR 2213689 (2007e:05012), https://doi.org/10.1090/S0002-9939-06-08224-4
  • [9] I. J. Good, Short proof of a conjecture by Dyson, J. Mathematical Phys. 11 (1970), 1884. MR 0258644 (41 #3290)
  • [10] Laurent Habsieger, Une $ q$-intégrale de Selberg et Askey, SIAM J. Math. Anal. 19 (1988), no. 6, 1475-1489 (French, with English summary). MR 965268 (89m:33002), https://doi.org/10.1137/0519111
  • [11] Kevin W. J. Kadell, A proof of Andrews' $ q$-Dyson conjecture for $ n=4$, Trans. Amer. Math. Soc. 290 (1985), no. 1, 127-144. MR 787958 (86f:33003), https://doi.org/10.2307/1999787
  • [12] Kevin W. J. Kadell, A proof of Askey's conjectured $ q$-analogue of Selberg's integral and a conjecture of Morris, SIAM J. Math. Anal. 19 (1988), no. 4, 969-986. MR 946655 (89h:33006b), https://doi.org/10.1137/0519067
  • [13] Kevin W. J. Kadell, Aomoto's machine and the Dyson constant term identity, Methods Appl. Anal. 5 (1998), no. 4, 335-350. MR 1669871 (2000m:33018)
  • [14] Kevin W. J. Kadell, A Dyson constant term orthogonality relation, J. Combin. Theory Ser. A 89 (2000), no. 2, 291-297. MR 1741011 (2001h:05021), https://doi.org/10.1006/jcta.1999.2928
  • [15] R. N. Karasev and F. V. Petrov, Partitions of nonzero elements of a finite field into pairs, Israel J. Math. 192 (2012), no. 1, 143-156. MR 3004078
  • [16] Gy. Károlyi, Note on a problem of Kadell, manuscript.
  • [17] Gy. Károlyi, A. Lascoux, and S. O. Warnaar, Constant term identities and Poincaré polynomials, Trans. Amer. Math. Soc., to appear.
  • [18] Gy. Károlyi, Z. L. Nagy, F. V. Petrov, and V. Volkov, A new approach to constant term identities and Selberg-type integrals, submitted.
  • [19] Michał Lasoń, A generalization of combinatorial Nullstellensatz, Electron. J. Combin. 17 (2010), no. 1, Note 32, 6. MR 2729390 (2012c:05352)
  • [20] I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), no. 6, 988-1007. MR 674768 (84h:17006a), https://doi.org/10.1137/0513070
  • [21] W. G. Morris, Constant term identities for finite and affine root systems, Ph.D. thesis, University of Wisconsin, Madison, 1982. MR 2631899
  • [22] Richard P. Stanley, The $ q$-Dyson conjecture, generalized exponents, and the internal product of Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 81-94. MR 777696 (86i:22026), https://doi.org/10.1090/conm/034/777696
  • [23] Richard P. Stanley, The stable behavior of some characters of $ {\rm SL}(n,{\bf C})$, Linear and Multilinear Algebra 16 (1984), no. 1-4, 3-27. MR 768993 (86e:22025), https://doi.org/10.1080/03081088408817606
  • [24] John R. Stembridge, A short proof of Macdonald's conjecture for the root systems of type $ A$, Proc. Amer. Math. Soc. 102 (1988), no. 4, 777-786. MR 934842 (89e:11062), https://doi.org/10.2307/2047309
  • [25] Kenneth G. Wilson, Proof of a conjecture by Dyson, J. Mathematical Phys. 3 (1962), 1040-1043. MR 0144627 (26 #2170)
  • [26] Doron Zeilberger, A Stembridge-Stanton style elementary proof of the Habsieger-Kadell $ q$-Morris identity, Discrete Math. 79 (1990), no. 3, 313-322. MR 1044230 (91b:05024), https://doi.org/10.1016/0012-365X(90)90338-I
  • [27] Doron Zeilberger and David M. Bressoud, A proof of Andrews' $ q$-Dyson conjecture, Discrete Math. 54 (1985), no. 2, 201-224. MR 791661 (87f:05015), https://doi.org/10.1016/0012-365X(85)90081-0

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05A19, 05A30, 33D05, 33D60

Retrieve articles in all journals with MSC (2010): 05A19, 05A30, 33D05, 33D60


Additional Information

Gyula Károlyi
Affiliation: School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4072, Australia
Address at time of publication: Institute of Mathematics, Eötvös University, Pázmány P. sétány 1/c, Budapest, 1117 Hungary
Email: karolyi@cs.elte.hu

Zoltán Lóránt Nagy
Affiliation: Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13–15, Budapest, 1053 Hungary
Email: nagyzoltanlorant@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12041-7
Keywords: Constant term identities, Laurent polynomials, Dyson's conjecture, Combinatorial Nullstellensatz
Received by editor(s): March 26, 2012
Received by editor(s) in revised form: August 16, 2012, and September 26, 2012
Published electronically: May 28, 2014
Additional Notes: This research was supported by the Australian Research Council, by ERC Advanced Research Grant No. 267165, and by Hungarian National Scientific Research Funds (OTKA) Grants 67676 and 81310
Communicated by: Jim Haglund
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society