Back to the main page



Description of the program:

The program is implemented in Borland C++ Builder for Microsoft Windows. It reads a file defining a quiver with relations and constructs the corresponding algebra. It computes the bases and finds the diagrams of indecomposable projective modules over the algebra. The main objective of the program is to compute the minimal projective resolutions of simple modules. For now, the program works for most dihedral and some semi-dihedral algebras. For every simple module Si the algorithm tries to construct a bicomplex such that its total complex gives the minimal projective resolution of Si.

The program window has 8 buttons and a text output area. They are used as follows:

Input data:

To read a data file, press the button Define quiver. Let us describe the input file format using the example of an algebra of the family D(3L). These algebras are defined by the following quiver with relations (k>1, s>1):

We write the composition from the right to the left. Let us give an example of an input file, the file d3l_3_2.txt .  It describes the dihedral algebra of the family D(3L) with the parameters k=3, s=2. We replace \alpha by a, \beta by b etc.


The input quiver with relations must be described as follows:

Output data:

When the file d3l_3_2.txt is loaded, the program window can look like the following:

The text in the window gives the bases of the indecomposable projective modules Pi and their dimensions. Inside the program, the basis elements of P1, P2, P3 are denoted as v, u, w with indexes. Their monomial representation is also given.

Press the Diagrams button to see the diagrams of Pi's:

In mathematical notation, these diagrams (for any parameters) can be written as follows:

Press the Bicomplex S0 button to see the bicomplex for S0 (its total complex gives the minimal projective resolution of the simple module S0 ):

In mathematical notation, this bicomplex can be written as follows (we write it in general, for any parameters, transpose and put the minus signs everywhere in the odd strings to get an anti-commutative bicomplex):

Back to the main page