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Does the Leclerc-Thibon involution exchange vertex operators of the first and second type?

This question is about $U_q ( \hat{\mathfrak{sl}}_2 )$ representation theory. There is a notion of vertex operators $\Phi_{\pm }(z)$ of first and $\Psi_{\pm}(z)$ of the second type. They are defined …
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Does the Leclerc-Thibon involution exchange vertex operators of the first and second type?

I found the answer to the first question. Second and third questions remain. \begin{align} \Phi_+ (z) \rightarrow K^{-1/2} \Psi_+ (q^{-1} z) \\ \Phi_- (z) \rightarrow K^{1/2} \Psi_- (q^{-1} z) \end{a …
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