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In this paper, we study extension groups of determinantal modules over apreprojective algebra using the Auslander-Reiten translation of the quiverassociated with it. More precisely, based on the recent work given by Aizenbudand Lapid, we calculate the extension group of a sort of so-calleddeterminantal modules, which is an analog of quantum minors in quantumcoordinate rings. In particular, we give an equivalent combinatorial conditionwhen the product of two quantum minors (up to q-power rescaling) belongs to thedual canonical basis of quantum coordinate rings in the Dynkin case. Moregenerally, we can check the quasi-commuting condition for any two quantumcluster monomials with the seeds of quantum minors.

Determinantal modules over preprojective algebras and representations of Dynkin quivers