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The Ramanujan Journal

, Volume 48, Issue 3, pp 509–543 | Cite as

Leading terms of relations for standard modules of the affine Lie algebras \(C_{n}^{(1)}\)

  • Mirko Primc
  • Tomislav ŠikićEmail author
Article
  • 85 Downloads

Abstract

In this paper, we give a combinatorial parametrization of leading terms of defining relations for the vacuum level k standard modules for the affine Lie algebra of type \(C_{n}^{(1)}\). Using this parametrization, we conjecture colored Rogers–Ramanujan type combinatorial identities for \(n\ge 2\) and \(k\ge 2\); the identity in the case \(n=k=1\) is equivalent to one of Capparelli’s identities.

Keywords

Affine (Kac–Moody)Lie algebras Vertex operator algebras Integrable highest weight representations Combinatorial bases of standard modules Leading terms of defining relations Rogers–Ramanujan type identities 

Mathematics Subject Classification

17B67 17B69 05A19 

Notes

Acknowledgements

We thank Arne Meurman for many stimulating discussions and help in understanding the combinatorics of leading terms, and we thank Jim Lepowsky for many useful comments and suggestions. We also thank Shashank Kanade for letting us use his computer program which gave us further numerical evidence for our combinatorial conjecture.

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Authors and Affiliations

  1. 1.Faculty of Science, Department of MathematicsUniversity of ZagrebZagrebCroatia
  2. 2.Faculty of Electrical Engineering and Computing, Department of Applied MathematicsUniversity of ZagrebZagrebCroatia

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