Computing various objects of an algebra from the poset of torsion classes
Date:
Seminar Talk in OCAMI Algebra Seminar at Online
Based on my paper and my SageMath program,
- From the lattice of torsion classes to the posets of wide subcategories and ICE-closed subcategories
- The lattice of torsion classes in SageMath
Abstract
We show that, by using only the poset structure of torsion classes, we can construct various combinatorial objects associated with an artin algebra: the posets of several classes of subcategories of the module category (wide subcategories, ICE-closed subcategories, and hearts of intervals of torsion classes), and for the $\tau$-tilting finite case, Demonet-Iyama-Jasso’s simplicial complex of 2-term silting complexes. As an application, the poset of wide subcategories of a preprojective algebra coincides with the shard intersection order on the Coxeter group.
Since we only need the poset structure, these computations can be done using a computer. I will introduce a SageMath program I have been developing which computes the above objects. In particular, we can compute them for every representation-finite special biserial algebra by using Geuenich’s String Applet.