Kirillov-Reshetikhin modules and quantum $K$-matrices

From a quantum $K$-matrix of the fundamental representation, we construct one for the Kirillov-Reshetikhin module by fusion construction. Using the $\mathrm{ı}$crystal theory by the last author, we also obtain combinatorial $K$-matrices corresponding to the symmetric tensor representations of affine type $A$ for all quasi-split Satake diagrams.