Glasnik Matematicki, Vol. 57, No. 2 (2022), 161-184. \( \)

NEW PARTITION IDENTITIES FROM \(C^{(1)}_\ell\)-MODULES

Stefano Capparelli, Arne Meurman, Andrej Primc and Mirko Primc

Dipartimento SBAI, Università di Roma La Sapienza, Roma, Italy
e-mail:stefano.capparelli@uniroma1.it

Department of Mathematics, University of Lund, Box 118, 22100 Lund, Sweden
e-mail:arne.meurman@math.lu.se

Kersnikova 11, 1 000 Ljubljana, Slovenia
e-mail:aprimc@gmail.com

Faculty of Science, University of Zagreb, Zagreb, Croatia
e-mail:primc@math.hr

Abstract.   In this paper we conjecture combinatorial Rogers-Ramanu­jan type colored partition identities related to standard representations of the affine Lie algebra of type \(C^{(1)}_\ell\), \(\ell\geq2\), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.

2020 Mathematics Subject Classification.   05A17, 17B67

Key words and phrases.   Rogers-Ramanujan type identities, affine Lie algebras

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.57.2.01

References:

  1. K. Alladi, G. E. Andrews and B. Gordon, Refinements and generalizations of Capparelli's conjecture on partitions, J. Algebra 174 (1995), 636–658.
    MathSciNet    CrossRef

  2. G. E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 4082–4085.
    MathSciNet    CrossRef

  3. G. E. Andrews, The theory of partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2, Addison-Wesley, 1976.
    MathSciNet

  4. M. K. Bos, Coding the principal character formula for affine Kac-Moody Lie algebras, Math. Comp. 72 (2003), 2001–2012.
    MathSciNet    CrossRef

  5. K. Bringmann, C. Jennings-Shaffer and K. Mahlburg, Proofs and reductions of Kanade and Russell partition identities, J. Reine Angew. Math. 766 (2020), 109–135.
    MathSciNet    CrossRef

  6. S. Capparelli, On some representations of twisted affine Lie algebras and combinatorial identities, J. Algebra  154 (1993), 335–355.
    MathSciNet    CrossRef

  7. J. Dousse and I. Konan, Generalisations of Capparelli's and Primc's identities, I: Coloured Frobenius partitions and combinatorial proofs, Adv. Math. 408 (2022), Paper no. 108571, 70 pp.
    MathSciNet    CrossRef

  8. B. Feigin, R. Kedem, S. Loktev, T. Miwa and E  Mukhin, Combinatorics of the \(\widehat{\mathfrak sl}_2\) spaces of coinvariants, Transform. Groups 6 (2001), 25–52.
    MathSciNet    CrossRef

  9. E. Feigin, The PBW filtration, Represent. Theory 13 (2009), 165-181.
    MathSciNet    CrossRef

  10. E. Feigin, G. Fourier and P. Littelmann, PBW filtration and bases for symplectic Lie algebras, Int. Math. Res. Not. IMRN 24 (2011), 5760–5784.
    MathSciNet    CrossRef

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

  12. V. G. Kac, Infinite-dimensional Lie algebras, third edition. Cambridge University Press, Cambridge, 1990.
    MathSciNet    CrossRef

  13. S. Kanade and M. C. Russell, IdentityFinder and some new identities of Rogers-Ramanujan type, Exp. Math. 24 (2015), 419–423.
    MathSciNet    CrossRef

  14. J. Lepowsky, Lectures on Kac-Moody Lie algebras, Université Paris VI, Spring, 1978.

  15. J. Lepowsky and R. L. Wilson, The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities, Invent. Math. 77 (1984), 199–290; II: The case \(A_1^{(1)}\), principal gradation, Invent. Math. 79 (1985), 417–442.
    MathSciNet    CrossRef    MathSciNet    CrossRef

  16. A. Meurman and M. Primc, Annihilating fields of standard modules of \({\mathfrak sl}(2,\mathbb C)\,\widetilde{}\) and combinatorial identities, Mem. Amer. Math. Soc. 137 (1999), no. 652.
    MathSciNet    CrossRef

  17. A. Primc, https://github.com/aprimc/discretaly

  18. M. Primc and T.  Šikić, Combinatorial bases of basic modules for affine Lie algebras \(C_n^{(1)}\), J. Math. Phys. 57 (2016), 091701, 19 pp.
    MathSciNet    CrossRef

  19. M. Primc and T. Šikić, Leading terms of relations for standard modules of \(C_{n}^{(1)}\), Ramanujan J. 48 (2019), 509–543.
    MathSciNet    CrossRef

  20. G. Trupčević, Bases of standard modules for affine Lie algebras of type \(C_{\ell}^{(1)}\), Comm. Algebra 46 (2018), 3663–3673.
    MathSciNet    CrossRef
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