Determinantal modules over preprojective algebras and representations of Dynkin quivers

Yingjin Bi

Published 2021-11-22, updated 2022-05-02Version 2

In this paper, we study extension groups of modules over a preprojective algebra via the Auslander-Reiten translation of the quiver associated with it. More precisely, based on the recent work given by Aizenbud and Lapid, we give a description of extension groups of a preprojective algebra in terms of the AR translation and nilpotent matrices. As a result, 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.