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Pfaffian identities and Virasoro operators

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A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition \(\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _{2m})\), we show that, for \(k\ge 1\), \(L_{k}Q_{\lambda } = \sum ^{2m}_{i= 1}(\lambda _i-k)Q_{\lambda -2k\epsilon _i}\), where \(L_k\) is the Virasoro operator given as the quadratic form of free bosons. This main formula follows from the Plücker-like bilinear identity of Q-functions as Pfaffians: \(\sum ^{2m}_{i=2}(-1)^{i}\partial _1Q_{\lambda _1,\lambda _i}\partial _1Q_{\lambda _2, \ldots ,\widehat{\lambda _i},\ldots , \lambda _{2m}}=0\), where \(\partial _1=\partial /\partial t_1\). This bilinear identity must be explained in geometric words. We conjecture that the Hirota bilinear equations of the KdV hierarchy are derived from this bilinear identity.

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  1. 1.

    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

  2. 2.

    Hoffman, P.N., Humphreys, J.F.: Projective Representations of the Symmetric Groups. Oxford University Press, Oxford (1992)

  3. 3.

    Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS Kyoto Univ. 19, 943–1001 (1983)

  4. 4.

    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)

  5. 5.

    Wakimoto, M., Yamada, H.-F.: The Fock representations of the Virasoro algebra and the Hirota equations of the modified KP hierarchies. Hiroshima Math. J. 16, 427–441 (1986)

  6. 6.

    Yamada, H.-F.: Reduced Fock representation of the Virasoro algebra. In: Proceedings of the 35th Symposium on Algebraic Combinatorics, pp. 35–45 (2018)

  7. 7.

    Yamada, H.-F.: Unpublished notes (2005)

  8. 8.

    You, Y.: Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups. Adv. Ser. Math. Phys. 7, 449–464 (1989)

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The funding was provided by KAKENHI (Grant No. 17K05180).

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Correspondence to Kazuya Aokage.

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To the memory of Kiyosato Okamoto.

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Aokage, K., Shinkawa, E. & Yamada, H. Pfaffian identities and Virasoro operators. Lett Math Phys (2020). https://doi.org/10.1007/s11005-020-01265-1

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  • Q-functions
  • Virasoro operators
  • Bilinear identity
  • KdV hierarchy

Mathematics Subject Classification

  • 17B68
  • 05E10