Positivity properties for canonical bases of modified quantum affine ${\mathfrak{s}\mathfrak{l}}_n$

The positivity property for canonical bases asserts that the structure constants of the multiplication for the canonical basis are in ${\mathbb N}[v,v^{-1}]$\mathrm{.}Let$\mathbf U$bethequantumgroupover${\mathbb Q}(v)$ associated with a symmetric Cartan datum. The positivity property for the positive part ${\mathbf{U}}^{+}$ of $\mathbf{U}$ was proved by Lusztig. He conjectured that the positivity property holds for the modified form $\dot{\mathbf{U}}$ of $\mathbf{U}$. In this paper, we prove that the structure constants for the canonical basis of $\dot{\mathbf{U}}({\widehat{\mathfrak{s}\mathfrak{l}}}_n)$ coincide with certain structure constants for the canonical basis of ${\mathbf U}(\widehat{\frak{sl}}_N)^+$for$n<N$. In particular, the positivity property for $\dot{\mathbf{U}}({\widehat{\mathfrak{s}\mathfrak{l}}}_n)$ follows from the positivity property for $\mathbf{U}({\widehat{\mathfrak{s}\mathfrak{l}}}_N{)}^{+}$.