computing cohomology groups and character tables (CGT project ideas)

[Aniket Joshi]

Hello, 

I have several ideas in computational group theory that seem doable. I am wondering if these have been considered before and if they are in the scope of GSoC for Sympy. I have worked with GAP and in particular used it to compute cohomology groups and character tables. In general the goal could be to implement several algorithms in the chapter of Holt's book on representation theory, and cohomology groups(chapter 7). Following are some ideas:

1. Computing first and second cohomology  groups of finitely presented groups over a finite field or the integers
2. Compute character tables of finite groups (using Burnside-Dixon-Schneider algorithm)
3. Test irreducibility of given representation (using Meataxe algorithm)

4. Adding classes on finite groups based on the ATLAS database of finite groups (http://brauer.maths.qmul.ac.uk/Atlas/v3/)

Since any of these ideas don't appear on the GSoC ideas page, please could you give me some feedback if they would be of general interest and can be considered as part of a GSoC project. 

Best, 
Aniket

[Aaron Meurer]

Hi.

Can you show an example of what this sort of thing looks like in GAP,
so we can get an idea of what it might look like in SymPy?

Aaron Meurer

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[Aniket Joshi]

Hello Aaron, 

Thanks for your interest. I am giving some examples below:

1. The following example computes the fourth integral cohomology of the Mathieu group M24:

 gap> GroupCohomology(MathieuGroup(24),4);

[ 4, 3 ]

2. Similarly the following example computes the second integral homology of the alternative group:

gap>G = AlternatingGroup(5) ;

Alt( [ 1 .. 5 ] ) 

gap> GroupHomology(G,2);

[2]

3.  The following example print the character table of S_4

gap> g:= SymmetricGroup( 4 );
Sym( [ 1 .. 4 ] )

gap>Charactertable(g);

4. Some of the character table operations offered by GAP are taking direct product of character tables, character table of a quotient, finding scalar product of characters in the same character table etc. 

Best, 
Aniket

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Aniket S Joshi

4th year B.S.-M.S. Physics Dual Degree 

Roll no. PH11B001

Department of Physics, IIT Madras