Elliptic determinant evaluations and the Macdonald identities for affine root systems

Compositio Mathematica

Volumn 142, Issue 4, 2006, Pages 937-961

  Rosengren, Hjalmar ^a   Schlosser, Michael ^b  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

^b UNIVERSITY OF VIENNA   (Austria)

Author keywords

Affine root systems; Determinants; Elliptic functions; Macdonald identities; Theta functions; Weyl denominator formula

Indexed keywords

EID: 33845302952     PISSN: 0010437X     EISSN: None     Source Type: Journal    
DOI: 10.1112/S0010437X0600203X     Document Type: Article

References (37)

1. DS01 J. F. van Diejen and V. P. Spiridonov, Elliptic Selberg integrals, Internat. Math. Res. Notices 20 (2001), 1083-1110.
2. Dys72. F. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652.
3. FK99 H. M. Farkas and I. Kra, On the quintuple product identity, Proc. Amer. Math. Soc. 127 (1999), 771-778.
4. Fay73 J. D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, vol. 352 (Springer, Berlin, 1973).
5. Fro82 F. G. Frobenius, Über die elliptischen Funktionen zweiter Art, J. reine angew. Math. 93 (1882), 53-68.
6. FS77 F. G. Frobenins and L. Stickelberger, Zur Theorie der elliptischen Functionen, J. reine angew. Math. 83 (1877), 175-179.
7. 1-transformations, in Special functions, q-series and related topics, eds M. E. H. Ismail, D. R. Masson and M. Rahman (American Mathematical Society, Providence, RI, 1997), 83-90.
8. Has97 K. Hasegawa, Ruijsenaars' commuting difference operators as commuting transfer matrices, Comm. Math. Phys. 187 (1997), 289-325.
9. Hir83 M. D. Hirschhorn, A simple proof of an identity of Ramanujan, J. Aust. Math. Soc. Ser. A 34 (1983), 31-35.
10. Hir00 M. D. Hirschhorn, An identity of Ramanujan, and applications, in q-Series from a contemporary perspective, eds M. E. H. Ismail and D. W. Stanton, Contemporary Mathematics, vol. 254 (American Mathematical Society, Providence, RI, 2000), 229-234.
11. Kac90 V. Kac, Infinite-dimensional Lie algebras, 3rd edn (Cambridge University Press, Cambridge, 1990).
12. KN03 Y. Kajihara and M. Noumi, Multiple elliptic hypergeometric series. An approach from the Cauchy determinant, Indag. Math. (N.S.) 14 (2003), 395-421.
13. Kra95 C. Krattenthaler, The major counting of nonintersecting lattice paths and generating functions for tableaux, Mem. Amer. Math. Soc. 115 (1995), no. 552.
14. Kra99 C. Krattenthaler, Advanced determinant calculus, Sem. Lothar. Combin. 42 (1999), B42q.
15. Kra05 C. Krattenthaler, Advanced determinant calculus: a complement, Linear Algebra Appl. 411 (2005), 68-166.
16. Mac72 I. G. Macdonald, Affine root systems and Dedekind's η-function, Invent. Math. 15 (1972), 91-143.
17. Oka04 S. Okada, An elliptic generalization of Schur's Pfaffian identity, Adv. Math., to appear.
18. Rai03 E. M. Rains, Transformations of elliptic hypergeometric integrals, Preprint (2003), math.QA/0309252.
19. Rai05 E. M. Rains, Recurrences for elliptic hypergeometric integrals, in Elliptic integrable systems, eds M. Noumi and K. Takasaki, Rokko Lectures in Mathematics, vol. 18 (Kobe University, 2005), 183-199.
20. Ros01 H. Rosengren, A proof of a multivariable elliptic summation formula conjectured by Warnaar, in q-Series with applications to combinatorics, number theory, and physics, eds B. C. Berndt and K. Ono, Contemporary Mathematics, vol. 291 (American Mathematical Society, Providence, RI, 2001), 193-202.
21. Ros04 H. Rosengren, Elliptic hypergeometric series on root systems, Adv. Math. 181 (2004), 417-447.
22. Ros05 H. Rosengren, Sums of triangular numbers from the Frobenius determinant, Preprint (2005), math.NT/0504272.
23. RS03 H. Rosengren and M. Schlosser, Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations, Indag. Math. (N.S.) 14 (2003), 483-514.
24. Rui87 S. N. M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys. 110 (1987), 191-213.
25. r basic hypergeometric series, Ramannjan J. 1 (1997), 243-274.
26. r, Ramanujan J. 3 (1999), 405-461.
27. r, in q-Series from a contemporary perspective, eds M. E. H. Ismail and D. W. Stanton, Contemporary Mathematics, vol. 254 (American Mathematical Society, Providence, RI, 2000), 413-432.
28. Sch00b M. Schlosser, Summation theorems for multidimensional basic hypergeometric series by determinant evaluations, Discrete Math. 210 (2000), 151-169.
29. Spi03 V. Spiridonov, Theta hypergeometric integrals, Algebra i Analiz 15 (2003), 161-215.
30. Sta89 D. Stanton, An elementary approach to the Macdonald identities, in q-Series and partitions, ed. D. Stanton, IMA Volumes in Mathematics and its Applications, vol. 18 (Springer, New York, 1989), 139-149.
31. Sy151 J. J. Sylvester, On the relation between the minor determinants of linearly equivalent quadratic functions, Philosophical Magazine 1 (1851), 295-305.
32. TV97 V. Tarasov and A. Varchenko, Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997).
33. War02 S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), 479-502.
34. Wat29 G. N. Watson, Theorems stated by Ramanujan. VII: Theorems on continued fractions, J. London Math. Soc. 4 (1929), 39-48.
35. Web91 H. Weber, Elliptische Functionen und algebraische Zahlen (Vieweg & Sohn, Braunschweig, 1891).
36. WW96 E. T. Whittaker and G. N. Watson, A course of modern analysis, reprint of the 1927 fourth edition (Cambridge University Press, Cambridge, 1996).
37. Win69 L. Winquist, An elementary proof of p(11m + 6) = 0 (mod 11), J. Combin. Theory 6 (1969), 56-59.