Abstract
In recent work on the representation theory of vertex algebras related to the Virasoro minimal models M(2, p), Adamović and Milas discovered logarithmic analogues of (special cases of) the famous Dyson and Morris constant term identities. In this paper we show how the identities of Adamović and Milas arise naturally by differentiating as-yet-conjectural complex analogues of the constant term identities of Dyson and Morris. We also discuss the existence of complex and logarithmic constant term identities for arbitrary root systems, and in particular prove such identities for the root system G2.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Adamović, D., Milas, A.: On W-algebras associated to (2, p) minimal models and their representations. Int. Math. Res. Not. IMRN 20, 3896–3934 (2010)
Adamović, D., Milas, A.: On W-algebra extensions of (2, p) minimal models: p > 3. J. Algebra 344, 313–332 (2011)
Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
Apagodu, M., Zeilberger, D.: Multi-variable Zeilberger and Almkvist–Zeilberger algorithms and the sharpening of Wilf–Zeilberger theory. Adv. Appl. Math. 37, 139–152 (2006). http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/multiZ.html
Bailey, W.N.: Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 32. Stechert-Hafner, Inc., New York (1964)
Bressoud, D.M.: Proofs and Confirmations—The Story of the Alternating Sign Matrix Conjecture. Cambridge University Press, Cambridge (1999)
Bressoud, D.M., Goulden, I.P.: Constant term identities extending the q-Dyson theorem. Trans. Amer. Math. Soc. 291, 203–228 (1985)
Cherednik, I.: Double affine Hecke algebras and Macdonald’s conjectures. Ann. Math. (2) 141, 191–216 (1995)
Dyson, F.J.: Statistical theory of the energy levels of complex systems I. J. Math. Phys. 3, 140–156 (1962)
Forrester, P.J., Warnaar, S.O.: The importance of the Selberg integral. Bull. Amer. Math. Soc. (N.S.) 45, 489–534 (2008)
Gessel, I.M., Xin, G.: A short proof of the Zeilberger–Bressoud q-Dyson theorem. Proc. Amer. Math. Soc. 134, 2179–2187 (2006)
Gessel, I.M., Lv, L., Xin, G., Zhou, Y.: A unified elementary approach to the Dyson, Morris, Aomoto, and Forrester constant term identities. J. Combin. Theory Ser. A 115, 1417–1435 (2008)
Good, I.J.: Short proof of a conjecture by Dyson. J. Math. Phys. 11, 1884 (1970)
Gunson, J.: unpublished
Habsieger, L.: La q-conjecture de Macdonald���Morris pour G 2. C. R. Acad. Sci. Paris Sér. I Math. 303, 211–213 (1986)
Habsieger, L.: Une q-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19, 1475–1489 (1988)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9. Springer, New York (1978)
Kadell, K.W.J.: A proof of Askey’s conjectured q-analogue of Selberg’s integral and a conjecture of Morris. SIAM J. Math. Anal. 19, 969–986 (1988)
Kadell, K.W.J.: Aomoto’s machine and the Dyson constant term identity. Methods Appl. Anal. 5, 335–350 (1998)
Kaneko, J.: Forrester’s conjectured constant term identity II. Ann. Comb. 6, 383–397 (2002)
Károlyi, G., Nagy, Z.L.: A short proof of Andrews’ q-Dyson conjecture. Proc. Amer. Math. Soc., so appear
Knuth, D.E.: Overlapping Pfaffians. Electron. J. Combin. 3, 13 (1996) (Research Paper 5)
Krattenthaler, C.: Advanced determinant calculus. Sém. Lothar. Combin. 42, 67 (1999) (Art. B42q)
Opdam, E.M.: Some applications of hypergeometric shift operators. Invent. Math. 98, 1–18 (1989)
Macdonald, I.G.: Some conjectures for root systems. SIAM J. Math. Anal. 13, 988–1007 (1982)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)
Morris, W.G.: Constant term identities for finite and affine root systems: conjectures and theorems. Ph.D. Thesis, University of Wisconsin-Madison (1982)
Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B. A. K. Peters, Ltd., Wellesley (1996)
Selberg, A.: Bemerkninger om et multipelt integral. Norske Mat. Tidsskr. 26, 71–78 (1944)
Sills, A.V.: Disturbing the Dyson conjecture, in a generally GOOD way. J. Combin. Theory Ser. A 113, 1368–1380 (2006)
Sills, A.V.: Disturbing the q-Dyson conjecture. In: Tapas in Experimental Mathematics. Contemporary Mathematics, vol. 457, pp. 265–271. American Mathematical Society, Providence (2008)
Sills, A.V., Zeilberger, D.: Disturbing the Dyson conjecture (in a GOOD way). Experiment. Math. 15, 187–191 (2006)
Stanton, D.: Sign variations of the Macdonald identities. SIAM J. Math. Anal. 17, 1454–1460 (1986)
Stembridge, J.R.: A short proof of Macdonald’s conjecture for the root systems of type A. Proc. Amer. Math. Soc. 102, 777–786 (1988)
Stembridge, J.R.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83, 96–131 (1990)
Wilson, K.: Proof of a conjecture of Dyson. J. Math. Phys. 3, 1040–1043 (1962)
Zeilberger, D.: A combinatorial proof of Dyson’s conjecture. Discrete Math. 41, 317–321 (1982)
Zeilberger, D.: A proof of the G 2 case of Macdonald’s root system–Dyson conjecture. SIAM J. Math. Anal. 18, 880–883 (1987)
Zeilberger, D.: A Stembridge–Stanton style elementary proof of the Habsieger–Kadell q-Morris identity. Discrete Math. 79, 313–322 (1990)
Zeilberger, D., Bressoud, D.M.: A proof of Andrews’ q-Dyson conjecture. Discrete Math. 54, 201–224 (1985)
Acknowledgements
The authors have greatly benefited from Tony Guttmann’s and Vivien Challis’ expertise in numerical computations. The authors also thank Antun Milas and the anonymous referee for very helpful remarks, leading to the inclusion of Sect. 11.6.3. OW and WZ are supported by the Australian Research Council.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
To Jon
Communicated By David H. Bailey.
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this paper
Cite this paper
Chappell, T., Lascoux, A., Warnaar, S.O., Zudilin, W. (2013). Logarithmic and Complex Constant Term Identities. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_11
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7621-4_11
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7620-7
Online ISBN: 978-1-4614-7621-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)