Abstract
We consider the principal subspaces of certain level \(k\geqslant 1\) integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types D, E and F. Generalizing the approach of G. Georgiev we construct their quasi-particle bases. We use the bases to derive presentations of the principal subspaces, calculate their character formulae and find some new combinatorial identities.
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Notes
Note that the quasi-particles of color 1 in type \(E_7\) correspond, with respect to the aforementioned identification, to the quasi-particles of color 7 in type \(E_8\); see Fig. 1.
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Acknowledgements
The authors would like to thank Mirko Primc for useful discussions and support. Also, we would like to thank the anonymous referees for the valuable comments and suggestions which helped us to improve the manuscript. This work has been supported by Croatian Science Foundation under the project UIP-2019-04-8488. The first author is partially supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004).
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Butorac, M., Kožić, S. Principal subspaces for the affine Lie algebras in types D, E and F. J Algebr Comb 56, 1063–1096 (2022). https://doi.org/10.1007/s10801-022-01146-x
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DOI: https://doi.org/10.1007/s10801-022-01146-x
Keywords
- Principal subspaces
- Combinatorial bases
- Combinatorial identities
- Quasi-particles
- Vertex operator algebras
- Affine Lie algebras