The comultiplication of modified quantum affine $\frak{sl}_n$

Abstract: Let $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ be the modified quantum affine $\frak{sl}_n$ and let ${\bf U}(\widehat{\frak{sl}}_N)^+$ be the positive part of quantum affine $\frak{sl}_N$. Let $\dot{\mathbf{B}}(n)$ be the canonical basis of $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ and let $\mathbf{B}(N)^{\mathrm{ap}}$ be the canonical basis of ${\bf U}(\widehat{\frak{sl}}_N)^+$. It is proved in \cite{FS} that each structure constant for the multiplication with respect to $\dot{\mathbf{B}}(n)$ coincide with a certain structure constant for the multiplication with respect to $\mathbf{B}(N)^{\mathrm{ap}}$ for $n<N$. In this paper we use the theory of affine quantum Schur algebras to prove that the structure constants for the comultiplication with respect to $\dot{\mathbf{B}}(n)$ are determined by the structure constants for the comultiplication with respect to $\mathbf{B}(N)^{\mathrm{ap}}$ for $n<N$. In particular, the positivity property for the comultiplication of $\dot{\mathbf{U}}(\widehat{\frak{sl}}_n)$ follows from the positivity property for the comultiplication of ${\bf U}(\widehat{\frak{sl}}_N)^+$.