[GAP Forum] Fwd: [GAP Support] char Table and LLL

[John McDermott]

This was sent to the Support list and was intended for the Forum as  
well, but was mis-addressed:

Begin forwarded message:

> From: "Previtali Andrea" <Andrea.Previtali at uninsubria.it>
> Date: 3 March 2005 18:16:15 GMT
> To: <Thomas.Breuer at Team4.de>, <support at gap-system.org>,  
> <gap-forum at gap-system.org>
> Cc: Subject: [GAP Support] char Table and LLL
>
> Dear Thomas,
> I'm trying to get the character table of PSp(4,9) using your  
> suggestions but
> I get a weird answer:
>
> tbl:= rec();;
>     tbl.Size:=74;;
>     tbl.UnderlyingCharacteristic:= 0;;
>     tbl.SizesConjugacyClasses:= [ 1, 3321, 298890, 3280, 3280, 236160,
> 295200, 597780, 597780, 26900100, 478224, 478224, 478224, 478224,  
> 34432128,
> 265680, 265680, 5313600, 5313600, 10627200,
> 23911200, 597780, 597780, 21520080, 21520080, 53800200, 21254400,  
> 21254400,
> 478224, 478224, 17216064, 17216064, 21520080, 21520080, 34432128,  
> 23911200,
> 23911200,
> 47822400, 19128960, 19128960, 19128960, 19128960, 38257920, 38257920,
> 43040160, 43040160, 43040160, 43040160, 23911200, 23911200, 23911200,
> 23911200, 19128960,
> 19128960, 19128960, 19128960, 43040160, 43040160, 43040160, 43040160,
> 43040160, 43040160, 43040160, 43040160, 41990400, 41990400, 41990400,
> 41990400, 41990400,
> 41990400, 41990400, 41990400, 41990400, 41990400 ];;
>     tbl.OrdersClassRepresentatives:= [ 1, 2, 2, 3, 3, 3, 3, 4, 4, 4,  
> 5, 5, 5,
> 5, 5, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10,  
> 10, 12,
> 12, 12, 15, 15, 15, 15, 15, 15, 20, 20, 20,
> 20, 24, 24, 24, 24, 30, 30, 30, 30, 40, 40, 40, 40, 40, 40, 40, 40,  
> 41, 41,
> 41, 41, 41, 41, 41, 41, 41, 41 ];;
>     tbl.Identifier:= "PSp(4,9)";;  # for printing
>     ConvertToLibraryCharacterTableNC( tbl );;
>     vecs:=[
>      [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1, 1,
> 1, 1, 1,
>     1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
>     [ 369, -31, 9, 45, 45, 9, 0, 1, 9, 1, 9, -1, -1, 9, -1, 5, 5, -4,  
> -4, 5,
> 0, 1, 1, -1, -1, 1, 0, 0, 9, 9, -1, -1, -1, -1, -1, 1, 1, 0, 0, 0, 0,  
> 0, -1,
> -1, -1,
>     1, -1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, -1, -1, -1, -1, 1, 0,  
> 0, 0, 0,
> 0, 0, 0, 0, 0, 0 ],
>     [ 450, 50, 10, 45, 45, 0, 9, 10, 10, 2, 0, 0, 0, 0, 0, 5, 5, 5, 5,  
> -4, 1,
> 10, 10, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0,  
> 0, 0,
> 0, 0,
>     1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1,  
> -1, -1,
> -1, -1, -1, -1 ]
>
>     ]
>  ;;
>     LLL( tbl, vecs );
>
> PRODUCES THIS ANSWER   !!!!!!!
>
> rec( irreducibles := [  ], remainders := [ VirtualCharacter(  
> CharacterTable(
> "PSp(4,9)" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1,
> 1,
>           1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1, 1,
> 1,
>           1, 1, 1, 1 ] ), VirtualCharacter( CharacterTable( "PSp(4,9)"  
> ), [
> 369, -31, 9, 45, 45, 9, 0, 1, 9, 1, -1, 9, 9, -1, -1, 5, 5, -4, -4, 5,  
> 0, 1,
> 1,
>           -1, -1, 1, 0, 0, 9, 9, -1, -1, -1, -1, -1, 1, 1, 0, 0, 0, 0,  
> 0, -1,
> -1, 1, -1, 1, -1, 1, 1, 1, 1, 0, 0, 0, 0, -1, 1, 1, 1, -1, -1, 1, -1,  
> 0, 0,
> 0,
>           0, 0, 0, 0, 0, 0, 0 ] ), VirtualCharacter( CharacterTable(
> "PSp(4,9)" ), [ 450, 50, 10, 45, 45, 0, 9, 10, 10, 2, 0, 0, 0, 0, 0,  
> 5, 5, 5,
> 5, -4, 1,
>           10, 10, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0,  
> 0, 0, 0,
> 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1,  
> -1,
> -1,
>           -1, -1, -1, -1, -1, -1 ] ), VirtualCharacter( CharacterTable(
> "PSp(4,9)" ), [ 820, 20, -20, 91, 91, 10, 10, 8, 0, 0, 0, 10, 10, 0,  
> 0, 11,
> 11, 2, 2,
>           2, -2, 10, 10, 0, 0, -2, 1, 1, 10, 10, 0, 0, 0, 0, 0, -1,  
> -1, 0, 1,
> 1, 1, 1, 0, 0, -2, 0, -2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0,  
> 0, 0,
> 0,
>           0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ], norms := [ 860803200/37,
> 860803200/37, 860803200/37, 860803200/37 ] )
>
> WHILE
>
>
> List([1..3],x->List([1..3],y->Sum(List([1..74],i- 
> >SizesConjugacyClasses(tbl)[
> i]*vecs[x][i]*vecs[y][i]))));
>
> PRoVIDES
>
>
> [ [ 1721606400, 0, 0 ], [ 0, 1721606400, 0 ], [ 0, 0, 1721606400 ] ]
>
> proving that vecs is a piece of an orthonormal basis for class  
> functions of
> PSp(4,9) (whose order is 1721606400)
>
> I'm really puzzled by the norm assigned to all v in vecs, in  
> particular to
> the one correspondign to the trivial character of G.
>
>
> If you have any hint why this happens, please let me know.
>
> bye
>
>
>
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> http://mail.gap-system.org/mailman/listinfo/support
>
>
--
John McDermott
Computing Officer & Purchase Officer
School of Computer Science
University of St Andrews
North Haugh, St Andrews, Fife
KY16 9SX

tel +44 1334 463252
mob +44 7941 507531