Determinantal modules over preprojective algebras and representations of Dynkin quivers

TL;DR

This paper investigates the extension groups of determinantal modules over preprojective algebras related to Dynkin quivers, providing combinatorial criteria for quantum minors' products to belong to the dual canonical basis.

Contribution

It introduces a method to compute extension groups of determinantal modules and characterizes when quantum minors' products are in the dual canonical basis for Dynkin quivers.

Findings

Calculated extension groups using Auslander-Reiten translation.

Established combinatorial conditions for quantum minors' products.

Provided criteria for quasi-commuting of quantum cluster monomials.

Abstract

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.