Computing various objects of an algebra from the poset of torsion classes
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
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.