  In this paper, we study extension groups of determinantal modules over a
preprojective algebra using the Auslander-Reiten translation of the quiver
associated with it. More precisely, based on the recent work given by Aizenbud
and Lapid, we calculate the extension group of a sort of so-called
determinantal modules, which is an analog of quantum minors in quantum
coordinate rings. In particular, we give an equivalent combinatorial condition
when the product of two quantum minors (up to q-power rescaling) belongs to the
dual canonical basis of quantum coordinate rings in the Dynkin case. More
generally, we can check the quasi-commuting condition for any two quantum
cluster monomials with the seeds of quantum minors.
