Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 052, 15 pages      arXiv:2307.12921      https://doi.org/10.3842/SIGMA.2025.052
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem

Michael J. Schlosser
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

Received December 26, 2024, in final form June 23, 2025; Published online July 06, 2025

Abstract
We introduce an algebra of elliptic commuting variables involving a base $q$, nome $p$, and $2r$ noncommuting variables. This algebra, which for $r=1$ reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of $r$ $q$-commuting variables. We present a multinomial theorem valid as an identity in this algebra, hereby extending the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a consistency relation satisfied by the elliptic weights and the Weierstraß type $\mathsf A$ elliptic partial fraction decomposition. From the elliptic multinomial theorem we obtain, by convolution, an identity equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel-Turaev ${}_{10}V_9$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, this derivation of Rosengren's $\mathsf A_r$ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity.

Key words: multinomial theorem; commutation relations; elliptic weights; elliptic hypergeometric series.

pdf (459 kb)   tex (23 kb)  

References

  1. Baker H.F., Abelian functions. Abel's theorem and the allied theory of theta functions, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1995.
  2. Bergman G.M., The diamond lemma for ring theory, Adv. Math. 29 (1978), 178-218.
  3. Coskun H., Gustafson R.A., Well-poised Macdonald functions $W_\lambda$ and Jackson coefficients $\omega_\lambda$ on $BC_n$, in Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, American Mathematical Society, Providence, RI, 2006, 127-155, arXiv:math.CO/0412153.
  4. Date E., Jimbo M., Kuniba A., Miwa T., Okado M., Exactly solvable SOS models: local height probabilities and theta function identities, Nuclear Phys. B 290 (1987), 231-273.
  5. Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, in The Arnold-Gelfand Mathematical Seminars, Birkhäuser, Boston, MA, 1997, 171-204.
  6. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia Math. Appl., Vol. 96, Cambridge University Press, Cambridge, 2004.
  7. Hoshi N., Katori M., Koornwinder T., Schlosser M., On an identity of Chaundy and Bullard. III. Basic and elliptic extensions, in Applications and $q$-Extensions of Hypergeometric Functions, Contemp. Math., Vol. 819, American Mathematical Society, Providence, RI, 2025, 233-254, arXiv:2304.10003.
  8. Jimbo M., Introduction to the Yang-Baxter equation, Internat. J. Modern Phys. A 4 (1989), 3759-3777.
  9. Lascoux A., Schützenberger M.-P., Le monoïde plaxique, in Noncommutative Structures in Algebra and Geometric Combinatorics, Quad. ''Ricerca Sci.'', Vol. 109, CNR, Rome, 1981, 129-156.
  10. Milne S.C., Multiple $q$-series and ${\rm U}(n)$ generalizations of Ramanujan's $_1\Psi_1$ sum, in Ramanujan Revisited, Academic Press, Boston, MA, 1988, 473-524.
  11. Rains E.M., $BC_n$-symmetric Abelian functions, Duke Math. J. 135 (2006), 99-180, arXiv:math.CO/0402113.
  12. Rosengren H., Elliptic hypergeometric series on root systems, Adv. Math. 181 (2004), 417-447, arXiv:math.CA/0207046.
  13. Rosengren H., Elliptic hypergeometric functions, in Lectures on Orthogonal Polynomials and Special Functions, London Math. Soc. Lecture Note Ser., Vol. 464, Cambridge University Press, Cambridge, 2021, 213-279, arXiv:1608.06161.
  14. Rosengren H., Warnaar S.O., Elliptic hypergeometric functions associated with root systems, in Encyclopedia of Special Functions: the Askey-Bateman Project, Cambridge University Press, Cambridge, 2020, 159-186, arXiv:1704.08406.
  15. Schlosser M., Elliptic enumeration of nonintersecting lattice paths, J. Combin. Theory Ser. A 114 (2007), 505-521, arXiv:math.CO/0602260.
  16. Schlosser M., A noncommutative weight-dependent generalization of the binomial theorem, Sém. Lothar. Combin. 81 (2020), B81j, 24 pages, arXiv:1106.2112.
  17. Spiridonov V.P., An elliptic incarnation of the Bailey chain, Int. Math. Res. Not. 2002 (2002), 1945-1977.
  18. Spiridonov V.P., Theta hypergeometric series, in Asymptotic Combinatorics with Application to Mathematical Physics, NATO Sci. Ser. II Math. Phys. Chem., Vol. 77, Kluwer, Dordrecht, 2002, 307-327, arXiv:math.CA/0303204.
  19. Spiridonov V.P., Elliptic hypergeometric terms, in Arithmetic and Galois Theories of Differential Equations, Sémin. Congr., Vol. 23, Soc. Math. France, Paris, 2011, 325-345, arXiv:1003.4491.
  20. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1996.

Previous article  Next article  Contents of Volume 21 (2025)