Abstract
We give an explicit recursive description of the Hilbert series and Gröbner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers–Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.
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References
Andrews, G.E.: On the proofs of the Rogers–Ramanujan identities. In: \(q\)-series and partitions (Minneapolis, MN, 1988),volume 18 of IMA Vol. Math. Appl., pp. 1–14. Springer, New York (1989)
Andrews, G.E.: The theory of partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1998). Reprint of the 1976 original
Andrews, G.E., Baxter, R.J.: A motivated proof of the Rogers–Ramanujan identities. Am. Math. Mon. 96(5), 401–409 (1989)
Bruschek, C., Mourtada, H., Schepers, J.: Arc spaces and the Rogers–Ramanujan identities. Ramanujan J. 30(1), 9–38 (2013)
Calinescu, C., Lepowsky, J., Milas, A.: Vertex-algebraic structure of the principal subspaces of certain \(A^{(1)}_1\)-modules. I. Level one case. Int. J. Math. 19(1), 71–92 (2008)
Capparelli, S., Lepowsky, J., Milas, A.: The Rogers–Ramanujan recursion and intertwining operators. Commun. Contemp. Math. 5(6), 947–966 (2003)
Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Undergraduate Texts in Mathematics, third edition. Springer, New York (2006)
Feigin, B.: Abelianization of the BGG resolution of representations of the Virasoro algebra. Funct. Anal. Appl. 45(4), 297–304 (2011)
Feigin, B., Stoyanovsky, A.: Functional models of the representations of current algebras, and semi-infinite Schubert cells. Funct. Anal. Appl. 28(1), 55–72 (1994)
Gorsky, E., Oblomkov, A., Rasmussen, J.: On stable Khovanov homology of torus knots. Exp. Math. 22(3), 265–281 (2013)
Ishii, S.: Jet schemes, arc spaces and the Nash problem. C. R. Math. Acad. Sci. Soc. R. Can. 29(1), 1–21 (2007)
Kanade, S.: On a Koszul complex related to the principal subspace of the basic vacuum module for \(A_1^{(1)}\). J. Pure Appl. Algebra 222(2), 323–339 (2018)
Paramonov, K.: Cores with distinct parts and bigraded Fibonacci numbers. Discret. Math. 341(4), 875–888 (2018)
van Ekeren, J., Heluani, R.: Chiral Homology of elliptic curves and Zhu’s algebra. arXiv preprint arXiv:1804.00017 (2018)
Acknowledgements
E.G. would like to thank Boris Feigin, Mikhail Bershtein, James Lepowsky, Kirill Paramonov, and Anne Schilling for useful discussions. O.K. thanks Eric Babson and Jésus de Loera for discussions in the initial stages of the project.
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E.G. acknowledges the Russian Academic Excellence Project 5-100 for its support. The work of E.G. and O.K. was supported by the NSF Grants DMS-1700814 and DMS-1559338. The work of E.G. in Sect. 6 was supported by the RSF Grant 16-11-10160. O.K. was also supported by the Ville, Kalle, and Yrjö Väisälä foundation of the Finnish Academy of Science and Letters.
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Bai, Y., Gorsky, E. & Kivinen, O. Quadratic ideals and Rogers–Ramanujan recursions. Ramanujan J 52, 67–89 (2020). https://doi.org/10.1007/s11139-018-0127-3
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DOI: https://doi.org/10.1007/s11139-018-0127-3