Algorithms in Representation Theory

This thesis deals with implementation of algorithm for computation of generator of almost split sequences ending at an indecomposable nonprojective module of path algebra over finite quiver. Algorithm is implemented in algebra system GAP (Groups, Algorithms, Programming) with additional package QPA (Quivers and Path Algebras).

Department of Algebra, Faculty of Mathematics and Physics, Charles University
Author: Marek Trunkát, Supervisor: RNDr. Jan Šťovíček, Ph.D., Department of Algebra

How to run

Prerequisites

1. Install GAP following this tutorial http://www.gap-system.org/Download/index.html
2. Install QPA package pas described at http://www.math.ntnu.no/~oyvinso/QPA/

Execution

1. Run GAP
2. Load QPA package via LoadPackage("QPA");
3. Copy whole src/code/algorithm.g to command line to register AlmostSplitSequence2 function along with other helper functions.
4. Now you can use this function to compute a generator of DTr(X) → E → X:

K := Rationals;
Q := Quiver(3, [[1, 2, "a"], [2, 3, "b"],[1, 3, "c"]]);
KQ := PathAlgebra(K,Q);
A := KQ;
matrices := [ ["a", [[1,0,0],[0,1,0]]], 
              ["b", [[0,1],[1,0],[0,1]]], 
              ["c", [[0,0],[1,0]]] ];
mX := RightModuleOverPathAlgebra(A,matrices);

E := AlmostSplitSequence2( mX );

To compute just an E module add following line:

E := Range(E[1]);