[GAP Forum] Characters of unitary groups

[dmitrii.pasechnik at cs.ox.ac.uk]

Dear Forum,
On Mon, Mar 25, 2019 at 10:29:23PM +0000, Iverson, Joseph W [MATH] wrote:
> 
> I would like to compute some stuff involving character tables of finite unitary groups, the smallest of which is GU(4,5). "Compute some stuff" means I need to be able to input a 4x4 matrix over GF(5^2) and get the value of any irreducible character as an output. Short of that, I would be happy with an ordered list of conjugacy class representatives for the table's columns. If I understand GAP terminology correctly, then I am asking for a character table with a stored group.
>  
> I have the impression that GAP's character table library should be able to help me. However:
> 1. I can't tell from the documentation how to ask it for the character table of GU(4,5).

IMHO there is no implementation in GAP of a generic character table for
the unitary groups, but a brute force approach is perhaps possible:

The following tells us that GU(4,5) has 1002 conjugacy classes.

gap> g:=GU(4,5);;
gap> t:=CharacterTable(g);;
gap> Length(ClassNames(t)); # took about 3 minutes on my Linux laptop
1002
gap> ConjugacyClasses(g); # long list of entries such as

  [ [ Z(5^2)^14, Z(5), Z(5^2)^10, Z(5) ], [ Z(5^2)^11, Z(5^2)^20, Z(5^2)^15, Z(5)^0 ], [ Z(5), Z(5^2)^11, Z(5^2)^13, Z(5^2)^11 ], 
      [ Z(5), Z(5^2)^15, Z(5^2)^23, Z(5^2)^8 ] ]^G, [ [ Z(5), Z(5^2)^22, Z(5^2)^2, Z(5^2)^22 ], [ Z(5^2)^3, Z(5)^2, Z(5^2)^7, Z(5^2)^16 ], 
      [ Z(5^2)^22, Z(5^2)^3, Z(5^2)^5, Z(5^2)^3 ], [ Z(5^2)^22, Z(5^2)^7, Z(5^2)^15, Z(5)^0 ] ]^G, 

gap> i:=Irr(t); # pretty quickly gives:
Error, reached the pre-set memory limit
(change it with the -o command line option) in
  rep[pnt] := 0; at /mnt/opt/Sage/sage-dev/local/share/gap/lib/oprtperm.gi:349 called from 
...
> 2. I'm not sure if the library can give me a table with a stored group.

it depends a lot on how bad the cyclotomic numbers will be.
It's worth trying, I think, on a machine with at least 8Gb (or more) of
RAM...

There are also ways in GAP to find some characters, and essentially drive the
construction of the complete table manually.

HTH
Dmitrii

> 
> If what I want is computationally infeasible (e.g. since GU(4,5) has order around 177 trillion), then that would be good to know, too.
> 
> All the best,
> Joey Iverson
> 
> Assistant Professor
> Department of Mathematics
> Iowa State University
> 
> 
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