# Programs

I’m interested in realizing and expressing various objects in the representation theory of algebra using computer, especially using SageMath.

## AR quiver calculator

A tool to deal with the Auslander-Reiten quiver of a category. You can input your translation quiver by your mouse and keyboard, save and load your translation quiver, and import the AR quiver from String Applet.

So far (in ver 0.2.1), you can

• Compute the dimension of Hom between two indecomposables and composition series of Hom functors Hom(-,Y) and Hom(X,-).
• (in the case of triangulated categories) Compute shifts, and list all objects which are self Ext^n-orthogonal for given values n.

In the near(?) future, I’ll add functions computing torsion classes, (tau)-tilting modules, cluster-tilting objects, and so on.

### Example

• The AR quiver of the cluster category of type A3.

• Computation of Hom (for the above category)

• Computation of shifts, and maximal Ext-orthogonals.

## The lattice of torsion classes in SageMath

tors_lattice.py below enables us to compute and construct various objects from the lattice of torsion classes of a τ-tilting finite algebra in SageMath. Internally, this defines a class FiniteTorsLattice, which is a subclass of a SageMath class for finite lattices: FiniteLatticePoset.

Once you input the lattice of torsion classes (e.g. using my StringApplet-to-SageMath-converter below), this program can compute (or construct) various objects which naturally arise in the representation theory of algebras in SageMath, such as the lattice of wide subcategories, the lattice of ICE-closed subcategories, the simplicial complex of support τ-tilting modules, and the number of indecomposable Ext-projectives of each torsion class or a heart of each interval, and so on.

## StringApplet to SageMath converter

This enables us to import the lattice of torsion classes in SageMath from String Applet. String Applet can compute the Hasse quiver of torsion classes for an inputted algebra (which should be representation-finite special biserial). This module makes a data which we can use to construct and study the lattice of torsion classes in SageMath (e.g. the kappa map below), from the TeX file exported by String Applet.

## Kappa maps for lattices

This adds methods to a Sage class FiniteLatticePoset which compute the (extended) kappa (dual) map. Using this, one can compute kappa maps for finite lattices in SageMath.

• GitHub Repository
• Manual
• Notes for representation theorists: I strongly recommend you to read this if you are working with representation theory of algebras, not just a lattice theory: this explains how the kappa map arises in the rep-theory, and demonstrates how to use it to study bricks and torsion classes.