Positivity VS Negativity of Canonical Bases

We provide examples for negativity of structure constants of the stably canonical basis of modified quantum $\mathfrak{gl}_n$ and an analogous basis of modified quantum coideal algebra of $\mathfrak{gl}_n$. In contrast, we construct the canonical basis of the modified quantum coideal algebra of $\mathfrak{sl}_n$, establish the positivity of its structure constants, the positivity with respect to a geometric bilinear form as well as the positivity of its action on the tensor powers of the natural representation. The matrix coefficients of the transfer map on these Schur algebras with respect to the canonical bases are shown to be positive. Formulas for canonical basis of the iSchur algebra of rank one are obtained.