Pfaffian identities and Virasoro operators

Kazuya Aokage, Eriko Shinkawa, Hiro Fumi Yamada

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition λ= (λ1, λ2, … , λ2m) , we show that, for k≥ 1 , LkQλ=∑i=12m(λi-k)Qλ-2kϵi, where Lk 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: ∑i=22m(-1)i∂1Qλ1,λi∂1Qλ2,…,λi^,…,λ2m=0, where ∂1= ∂/ ∂t1. 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.

Original languageEnglish
Pages (from-to)1381-1389
Number of pages9
JournalLetters in Mathematical Physics
Issue number6
Publication statusPublished - 2020 Jun 1
Externally publishedYes


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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