A formula for Schur Q -functions is presented which describes the action of the Virasoro operators. For a strict partition $$\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _{2m})$$ λ = ( λ 1 , λ 2 , … , λ 2 m ) , we show that, for $$k\ge 1$$ k ≥ 1 , $$L_{k}Q_{\lambda } = \sum ^{2m}_{i= 1}(\lambda _i-k)Q_{\lambda -2k\epsilon _i}$$ L k Q λ = ∑ i = 1 2 m ( λ i - k ) Q λ - 2 k ϵ i , where $$L_k$$ 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: $$\sum ^{2m}_{i=2}(-1)^{i}\partial _1Q_{\lambda _1,\lambda _i}\partial _1Q_{\lambda _2, \ldots ,\widehat{\lambda _i},\ldots , \lambda _{2m}}=0$$ ∑ i = 2 2 m ( - 1 ) i ∂ 1 Q λ 1 , λ i ∂ 1 Q λ 2 , … , λ i ^ , … , λ 2 m = 0 , where $$\partial _1=\partial /\partial t_1$$ ∂ 1 = ∂ / ∂ 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.

中文翻译：

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Pfaffian 身份和 Virasoro 运营商

给出了 Schur Q 函数的公式，它描述了 Virasoro 算子的作用。对于严格分区 $$\lambda =(\lambda _1,\lambda _2,\ldots,\lambda _{2m})$$ λ = ( λ 1 , λ 2 , … , λ 2 m ) ，我们证明，对于 $$k\ge 1$$ k ≥ 1 , $$L_{k}Q_{\lambda } = \sum ^{2m}_{i= 1}(\lambda _i-k)Q_{\lambda -2k \epsilon _i}$$ L k Q λ = ∑ i = 1 2 m ( λ i - k ) Q λ - 2 k ϵ i ，其中 $$L_k$$ L k 是 Virasoro 算子作为自由的二次形式给出玻色子。这个主要公式遵循作为 Pfaffians 的 Q 函数的类似 Plücker 的双线性恒等式： $$\sum ^{2m}_{i=2}(-1)^{i}\partial _1Q_{\lambda _1,\lambda _i}\partial _1Q_{\lambda _2, \ldots ,\widehat{\lambda _i},\ldots , \lambda _{2m}}=0$$ ∑ i = 2 2 m ( - 1 ) i ∂ 1 Q λ 1 , λ i ∂ 1 Q λ 2 , ... , λ i ^ , ... , λ 2 m = 0 ，其中 $$\partial _1=\partial /\partial t_1$$ ∂ 1 = ∂ / ∂ t 1 。这种双线性恒等式必须用几何词来解释。我们推测 KdV 层次结构的 Hirota 双线性方程是从这个双线性恒等式导出的。