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Macdonald identities and multidimensional theta-functions

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Abstract

A new proof of the combinatorial Macdonald identities is presented. It is shown that one may regard these identities as a decomposition of certain multidimensional theta-functions into infinite products. The proof is based on some analytical properties of theta-functions. It is briefly discussed how one can modify the proof in order to replace analytical arguments by formal ones involving only operations with formal series.

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References

  1. Yu. Bazlov, “The homology of graded infinite-dimensional Lie algebras in connection with Macdonald identity” (in this issue).

  2. A. M. Vershik, “A bijective proof of the Jacobi identity and reshapings of the Young diagrams,”Zap. Nauchn. Semin. LOMI,155, 3–6 (1986).

    Google Scholar 

  3. H. Garland and J. Lepowsky, “Lie algebra and the Macdonald-Kac formulas,”Invent. Math.,34, 37–76 (1976).

    Article  Google Scholar 

  4. Z. L. Leibenzon, “A simple combinatorial proof of the Jacobi identity and its generalizations,”Funkts. Anal. Prilozh.,20, 77–78 (1986).

    Google Scholar 

  5. J. Lepowsky, “Generalized Verma modules, loop space and Macdonald-type identities,”Ann. l'école Norm. Sup.,12, 169–234 (1979).

    Google Scholar 

  6. E. Looijenga, “Root systems and elliptic curves,”Invent. Math.,38, 17–32 (1976).

    Article  Google Scholar 

  7. G. Macdonald, “Affine root systems and Dedekind'sη-functions,”Invent. Math.,15, 91–143 (1972).

    Article  Google Scholar 

  8. D. Mumford,Tata Lectures on Theta, I, II, Progress in Math., Birkhauser28,43, (1983, 1984).

  9. D. B. Fuchs,Cohomologies of Infinite-Dimensional Lie Algebras [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  10. V. Kac,Infinite Dimensional Lie Algebras, Cambridge University Press (1985).

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Authors
  1. M. A. Vsemirnov
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Additional information

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 67–77.

The author is grateful to A. M. Vershik and Yu. M. Bazlov for their interest and helpful comments.

This research was supported by ISSEP (grant No. A96-1965).

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Vsemirnov, M.A. Macdonald identities and multidimensional theta-functions. J Math Sci 96, 3486–3492 (1999). https://doi.org/10.1007/BF02175826

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  • DOI: https://doi.org/10.1007/BF02175826

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