Computing various objects of an algebra fromthe poset of torsion classes (Demo)

Haruhisa Enomoto
26 July 2021, OCAMI Algebra Seminar
Graduate School of Science, Osaka Prefecture University

1Outline

About SageMath
Installation
Input the poset of torsion classes
Enjoy!

2

About SageMathWhat is SageMath?

A free open-source mathematics software system
Why SageMath?
• SageMath provides a framework for finite posets, finitelattices, simplicial complexes, and so on.
• We can construct them and compute various things, e.g.
the set of join-irreducibles in the lattice.
• SageMath can check whether two objects are isomorphic.

3tors_lattice

I’ve developed a SageMath program tors_lattice.py
1. Input the poset of torsion classes, then
2. This program can construct• the lattice of wide subcategories,• the simplicial complex ∆(Λ),

etc.

4

Installation

Step 1: Install SageMath• Install SageMath (ver ≥ 9.0) on your computer:
Download from the official cite
https://www.sagemath.org/index.html
and just install it!
• From now on, we’ll use SageMath Notebook
Another option:
You can use CoCalc without installing sage5

Step 2: Download my program and load it• Download my file fromhttps://haruhisa-enomoto.github.io/files/tors_lattice.py• Create a SageMath notebook in the same directory as you
download this file.
• Execute load("tors_lattice.py")
Then you can use it!

6Input the poset of torsion classes

Input your poset

First, input the poset of torsion classes of your algebra in
SageMath, and name it poset for example.
Ways to input posets
1. For path algebras and preprojective algebras of Dynkin
type, SageMath already has it! (later)
2. If you have a Hasse diagram, then you can input itmanually.
3. You can import it from Jan Geuenich’s String Applet using
my other program, next slide.

7

String Applet to SageMath converter
String Applet can calculate the poset of torsion classes of any
representation-fintie special biserial algebra.
You can import it in SageMath using my converter.
1. Input your algebra in String Applet,and show its sfi -Tilting quiver.
2. Export a latex file (e.g. data.tex) in your working directory.
3. Download
https://haruhisa-enomoto.github.io/files/converter.py

in your working directory
4. Load it in your notebook by load("converter.py")
5. Execute poset = Poset(SAtoSage("data.tex"))
8

Enjoy!

What’s next?
Now you have your poset. Then execute
tors = TorsLattice(poset)
Once you have done it, you can construct various things, e.g.
• tors.wide_lattice()the lattice of wide subcategories• tors.ice_lattice()the lattice of ICE-closed subcategories• tors.heart_poset()
the poset of torsion hearts• tors.s_tau_tilt_complex()
the simplicial complex ∆(Λ) of fi -tilting pairs
See Manual for the list of all things you can do.
9

Demo

Q: Dynkin quiver, ΠQ : its preprojective algebra

algebrakQΠQtors(−)

wide(−)∆(−)

Cambrian latticeNon-crossing partition

Cluster cpx

Weyl grp with weak ordershard intersection order(???)

All these objects are already in SageMath!
Let’s check the above table, and guess (???)!

10

Demo

Q: Dynkin quiver, ΠQ : its preprojective algebra

algebrakQΠQtors(−)

wide(−)∆(−)

Cambrian latticeNon-crossing partition

Cluster cpx

Weyl grp with weak ordershard intersection order

Coxeter cpx

Red are conjectured by SageMath experiment (then proved).

11

Demo: lattice properties for wide Λ

There’re lots of properties SageMath can check for lattices.
Conjecture (some have been confirmed)
If Λ is fi -tilting finite, then wide Λ is:• ranked (graded), with its rank function given by thenumber of simples• Rank-symmetric• Relatively complemented• (strongly) Sperner

12Conjecture of ICE-closed subcategories

Sakai’s Conjecture (not true...)The number of Hasse arrows in ICE Λ starting at C is equal to
the number of indecomposable Ext-projectives in C.
True for hereditary and Nakayama algebras.
Find a counterexample!

13

Links• The SageMath notebook used in this demo
• SageMath codes in my website• On SageMath• SageMath tutorial• Finite Coxeter groups• Finite posets• Finite lattices and semilattices• The lattice of torsion classes in SageMath• Download• Manual• String Applet to SageMath converter• Download• Manual14