Canonical bases of a coideal subalgebra in $U_q(\mathfrak{sl}_2)$

We consider tensor products of finite-dimensional representations of a coideal subalgebra in $U_{q}(\mathfrak{sl}_2)$. We present an explicit expression for the dual of the canonical bases through a diagrammatic presentation. We show that the decomposition of tensor products of dual canonical bases and the action of the coideal subalgebra have integral and positive properties. As an application, we consider the eigensystem of the generator of the coideal subalgebra on the dual canonical bases. We provide all the eigenvalues and obtain an explicit expression of the eigenfunction for the largest eigenvalue. The sum of the components of this eigenfunction is conjectured to be equal to the total number of arrangements of bishops with a certain symmetry.