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See the following example code snippets in GAP:

gap> M1:=[[0,0,1,0],[1,0,0,0],[0,-1,0,0],[1/4,1/4,1/4,1]];;
gap> M2:=[[0,0,-1,0],[0,-1,0,0],[1,0,0,0],[0,0,0,1]];;
gap> S:=AffineCrystGroup([M1,M2]);
<matrix group with 2 generators>
gap> Size(S);
infinity
gap> IrreducibleRepresentations(S);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 2nd choice method found for `IrreducibleRepresentations' on 1 arguments at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/methsel2.g:249 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:5
type 'quit;' to quit to outer loop
brk> Irr(S);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 4th choice method found for `ConjugacyClasses' on 1 arguments at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/methsel2.g:249 called from
ConjugacyClasses( G ) at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/ctbl.gi:1077 called from
OrdinaryCharacterTable( G ) at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/ctbl.gi:810 called from
Irr( G, 0 ) at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/ctbl.gi:792 called from
ErrorNoReturn( no_method_found ); at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/methsel2.g:249 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *errin*:1
type 'quit;' to quit to outer loop

It's well known that Space Groups Irreducible Representations and character table exist. Why can't these things be directly calculated in GAP?

Regards, HZ

Olexandr Konovalov
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Hongyi Zhao
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1 Answers1

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I am going to answer the question from the title. Space groups are infinite. They in general admit infinitely many irreducible representations, and can't in general be computed algorithmically.

Also, character tables are for finite groups.

Max Horn
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  • But, I also noticed the following description here: The program REPRES computes the irreducible representations for all space groups G. For a given space group G and a k-vector the corresponding little group L is calculated, the irreducible representations of L are determined as well as induction matrices. Thus, the full-group representations are obtained. – Hongyi Zhao May 17 '22 at 06:24
  • Your code is asking for all irreducible representations of those groups. That page you link to surely is only about computing "crystallographic" / "space group" representations. These are a special kind of representation, and a special kind of group, so for those algorithms exist. But that's not what you are asking GAP to compute, so it can't comply. – Max Horn May 17 '22 at 07:06
  • As I told you before, you can use the GAP package cryst and crystcat to get this kind of data, see https://www.gap-system.org/Manuals/pkg/crystcat/htm/CHAP001.htm – Max Horn May 17 '22 at 07:09