Skip to main content

Looking for a New Version of Gordon’s Identities

Abstract

We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in Gordon’s identities, which are a generalization of Rogers–Ramanujan identities. Using this approach and differential ideals, we conjecture a family of partition identities which extend Gordon’s identities. This family is indexed by \(r\ge 2.\) We prove the conjecture for \(r=2\) and \(r=3.\)

This is a preview of subscription content, access via your institution.

References

  1. 1.

    P. Afsharijoo, Looking for a new member of Gordon’s identities, from algebraic geometry to combinatorics through partitions, Ph.D thesis, Université Paris Diderot, defended in May 2019.

  2. 2.

    P. Afsharijoo, H. Mourtada, Partition identities and application to finite dimensional Gröbner basis and viceversa. Arc Schemes and Singularities, World Scientific Publishing, pp. 145-161 (2020).

  3. 3.

    G.E. Andrews. The theory of partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original.

  4. 4.

    G. E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, American Math. Monthly 96 (1989), 401-409.

    MathSciNet  Article  Google Scholar 

  5. 5.

    C. Bruschek, H. Mourtada, J. Schepers, Arc spaces and Rogers-Ramanujan identities, The Ramanujan Journal: Volume 30, Issue 1 (2013), Page 9-38.

    MathSciNet  Article  Google Scholar 

  6. 6.

    C. Bruschek, H. Mourtada, J. Schepers, Arc spaces and Rogers-Ramanujan Identities, Discrete Mathematics and Theoretical Computer Science Proceedings, FPSAC (2011), 211- 220.

  7. 7.

    B. Feigin, S. Loktev, On finitization of the Gordon identities. Funct Anal. and Appl., vol. 35 (2001), no. 1.

  8. 8.

    S. Capparelli, J. Lepowsky, A. Milas, The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators. The Ramanujan Journal: 12:379-397,2006.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Y. Bai, E. Gorsky, O.Kivinen, Quadratic ideals and Rogers-Ramanujan recursions. The Ramanujan Journal: 52, 67-89 (2020).

    MathSciNet  Article  Google Scholar 

  10. 10.

    B. Gordon. A combinatorial generalization of the Rogers-Ramanujan identities. Amer. J. Math. 83 (1961), 393-399.

    MathSciNet  Article  Google Scholar 

  11. 11.

    G.-M Greulel and G. Pfister. A singular introduction to commutative algebra. Springer-Verlag, Berlin, 2002. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, With 1 CD-ROM (Windows, Macintosh, and UNIX).

  12. 12.

    J. Lepowsky, M. Zhu, A motivated proof of Gordon’s identities, The Ramanujan Journal: 29 (2012), no. 1-3, 199-211.

    MathSciNet  Article  Google Scholar 

  13. 13.

    H. Mourtada, Jet schemes of rational double point surface singularities, Valuation Theory in Interaction, EMS Ser. Congr. Rep., Eur. Math. Soc., Sept. 2014, pp: 373-388.

  14. 14.

    H. Mourtada, Jet schemes of complex plane branches and equisingularity. Annales de l’Institut Fourier, Tome 61, numéro 6 (2011), p. 2313-2336.

Download references

Acknowledgements

I would like to express my deep gratitude to Hussein Mourtada, my Ph.D advisor, for suggesting me this project and for his constant help. I am very thankful to Bernard Teissier for his help and his corrections of the earlier version of this paper. I also would like to thank Jehanne Dousse and Frédéric Jouhet for showing interest in this work and for giving me the opportunity to talk about it in their seminar. I also would like to thank the two anonymous reviewers for their useful suggestions and comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pooneh Afsharijoo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Ken Ono

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Afsharijoo, P. Looking for a New Version of Gordon’s Identities. Ann. Comb. 25, 543–571 (2021). https://doi.org/10.1007/s00026-021-00530-x

Download citation

Keywords

  • Gordon’s identities
  • Space of arcs
  • Hilbert series

Mathematics Subject Classification

  • 05A17
  • 12H05
  • 13D40
  • 13P10