Rigid modules and ICE-closed subcategories in quiver representations

Comment

In the previous paper, a subcategory of a module category closed under kernels, extensions and images naturally arises. We consider the dual version, Image-Cokernel-Extension closed subcategories (subcategory closed under taking Images, Cokernels and Extension), in the quiver representation. We found that this ICE-closed subcategories are in bijection with rigid modules (modules without self-extensions), which generalizes a bijection between torsion classes and support tilting modules due to Ingalls-Thomas.

We also show that the number of ICE-closed subcats only depends on the underlying graph of Dynkin quiver. This paper contains an explicit formula for the number of ICE-closed subcats for each Dynkin type. For type A, this number coincides with the large Schroeder number.

(Later I noticed that the main theorem can be better understood by considering exceptional sequences, and I added it.)