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 \(r=3.\)
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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.
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).
G.E. Andrews. The theory of partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original.
G. E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, American Math. Monthly 96 (1989), 401-409.
C. Bruschek, H. Mourtada, J. Schepers, Arc spaces and Rogers-Ramanujan identities, The Ramanujan Journal: Volume 30, Issue 1 (2013), Page 9-38.
C. Bruschek, H. Mourtada, J. Schepers, Arc spaces and Rogers-Ramanujan Identities, Discrete Mathematics and Theoretical Computer Science Proceedings, FPSAC (2011), 211- 220.
B. Feigin, S. Loktev, On finitization of the Gordon identities. Funct Anal. and Appl., vol. 35 (2001), no. 1.
S. Capparelli, J. Lepowsky, A. Milas, The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators. The Ramanujan Journal: 12:379-397,2006.
Y. Bai, E. Gorsky, O.Kivinen, Quadratic ideals and Rogers-Ramanujan recursions. The Ramanujan Journal: 52, 67-89 (2020).
B. Gordon. A combinatorial generalization of the Rogers-Ramanujan identities. Amer. J. Math. 83 (1961), 393-399.
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).
J. Lepowsky, M. Zhu, A motivated proof of Gordon’s identities, The Ramanujan Journal: 29 (2012), no. 1-3, 199-211.
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.
H. Mourtada, Jet schemes of complex plane branches and equisingularity. Annales de l’Institut Fourier, Tome 61, numéro 6 (2011), p. 2313-2336.
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.
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Communicated by Ken Ono
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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
- Gordon’s identities
- Space of arcs
- Hilbert series
Mathematics Subject Classification