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Pfaffian identities and Virasoro operators

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Abstract

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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Funding

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

Author information

Correspondence to Kazuya Aokage.

Additional information

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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Keywords

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

Mathematics Subject Classification

  • 17B68
  • 05E10