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 to be intertwiners $$\Phi(z): V(\Lambda_i) \rightarrow V(\Lambda_{1-i}) \otimes V_z $$ $$\Psi(z): V(\Lambda_i) \rightarrow V_z \otimes V(\Lambda_{1-i}) $$ here $V_z$ -- evaluation module corresponding to two-dimensional standard representation. $V(\Lambda_0)$ and $V(\Lambda_1)$ are the only two integrable representations of $U_q ( \hat{\mathfrak{sl}}_2 )$. Also let us define $\Phi_{\pm} (z): V(\Lambda_i) \rightarrow V(\Lambda_{1-i})$ by formula $\Phi(z) = \Phi_{+}(z) \otimes v_{+} + \Phi_{-}(z) \otimes v_-$ (here $v_{\pm}$ -- standard basis of two dimensional representation). The details can be found in a classical textbook by Jimbo and Miwa (chapter 6).

There is a paper by Stern. It studies $V(\Lambda_i)$ and one kind of vertex operators in terms of semi-infinite power of evaluation representation. I believe it is the second kind. In Stern's notation semi-infinite product is infinite to the right and truncated to the left. So it is easy to tensor with another evaluation representation on the left.

- There is an involution defined be Leclerc and Thibon (one can find definition here, section 3). The involution here is defined again in terms of semi-infinite tensor power of evaluation representation. $$u_{i_1} \wedge_q u_{i_2} \wedge_q \dots \wedge_q u_{k_i} \wedge_q u_{k_{i+1}} \wedge_q \dots \rightarrow (-1)^{{k}\choose{2} } q^{\alpha_{n,k}(I)} u_{i_k} \wedge_q u_{i_{k-1}} \wedge_q \dots \wedge_q u_{i_1} \wedge_q u_{i_{i+1}} \wedge_q \dots$$ This involution is antilinear. It means that it sends $q \rightarrow q^{-1}$.

Consider $\Psi_{\pm} (z)$ conjugated by involution. It is natural to conjecture, that this operator is $\Phi_{\pm}(z)$ because the conjugation "exchange left and right multiplication". But maybe it is true up to some renormalization... I tried to verify some defining relation (from Jimbo-Miwa) to check this and did not succeed. I do not have an idea, what is precisely the conjugated $\Psi$.

By the way, I suppose, that if these conjecture true, then it must be published (all my references are from the nineties). But I did not find anything.

Questions

- What is conjugated by Leclerc-Thibon involution to $\Psi_{\pm}(z)$?
- Do you know any reference for this?
- Is there any good way to find this (using $R$-matrix or semi-infinite wedge)?