TauTilting

Enumeration of sequences of support 𝜏-tilting modules with prescribed support-rank.

This here is research work that will only make sense to the few people in the world who've both studied 𝜏-tilting theory, and have a decent understanding of GAP/QPA, Go, and SymPy. We're computing sequences enumerating the number of support 𝜏-tilting modules with prescribed support defect over sequences of algebras, and fitting generating functions to the resulting data.

If this work looks relevant to your own or there are any questions, please contact the author.

How To

There needs to be a folder "MyResultFolder". This folder needs to contain a "comp.gap" file, containing a single function definition. This function is called "comp". It takes two positive integer arguments, n and K, and returns an algebra object. n denotes the number of vertices. K denotes the power of the arrow ideal taken as quotient, usually. Example:

 comp := function(n, K)
     local A, Q, orientation;
     orientation = [];
     for i in [1..n-1] do
         Add(orientation, "r");
     od;
     Q := DynkinQuiver("A", n, orientation);
     A := TruncatedPathAlgebra(Rationals, Q, K);

     return A;
 end;

Open GAP in this folder, and call the following:

 Read("compute_data.gap");

 ComputeAlgebrasModK("data/MyResultFolder", n_start, n_end, K);

The code will compute all algebras, and indecomposable modules (along with whether the direct sum of any two modules is \tau-rigid or not) for the algebras given by comp(n, K), where n ranges from n_start to n_end, inclusive. The data will be stored as alg_n.gap and as data_n.json, in the .JSON format. The data_n.json contains the alg_n.gap data, and is human readable.

To use this data to generate a table enumerating the number of support \tau-tilting modules with prescribed support rank for a given sequence of algebras, call

ComputeData("data/MyResultFolder", number_threads, granularity, false)

in main.go, and run the code via go run .

The resulting data can then be further analyzed using the tools/scripts in the analysis/ folder. Creating a Solution object with a given set of data and a hypothesis attached to it, allows checking whether the enumerated values can be approximated using a family of generating functions or not.