[GAP Forum] A bug in CharacterTable("2.A11.2") mod 3

[Lukas Maas]

Dear All,

the discrepancy that Shunsuke Tsuchioka noticed comes from a mistake in the paper by Morris and Yaseen. 
This was discussed by Thomas Breuer, Juergen Mueller and me some days ago, 
unfortunately in the wrong mailing list. So here is the full correspondence.

Best wishes, 
Lukas


On Feb 28, 2011, at 12:34 PM, Shunsuke Tsuchioka wrote:

> Hi,
> 
> I believe I found a bug in CharacterTable.
> 
> It is concluded that there exist 3-modular spin 
> irreducible representations of dimension 1440-144=1296 
> of the Schur cover of the symmetric group of degree 11 
> in the paper:
> 
> Morris, A. O.(4-WALA); Yaseen, A. K.(4-WALA)
> Decomposition matrices for spin characters of symmetric groups.
> Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 1-2, 145
> 
> However, in gap, the command:
> 
> c:=CharacterTable("2.A11.2") mod 3;
> Display(c);
> 
> displays the following infomation 
> on the spin irreducible representations:
> 
> X.28      32   -32   .  .  .  .  .  -8   8  2  -2  4  -4  .   .  -1   1   .
> X.29     144  -144   .  .  .  .  . -16  16 -1   1  4  -4  .   .   1  -1   .
> X.30     144  -144   .  .  .  .  . -16  16 -1   1  4  -4  .   .   1  -1   .
> X.31     528  -528   .  .  .  .  . -12  12 -2   2 -4   4  .   .   .   .   .
> X.32     640  -640   .  .  .  .  .  20 -20  .   . -4   4  .   .   2  -2   .
> X.33    1440 -1440   .  .  .  .  .  20 -20  .   . -2   2  .   .  -1   1   .
> X.34    1440 -1440   .  .  .  .  .  20 -20  .   . -2   2  .   .  -1   1   .
> 
> I believe X.33 and X.34 are not irreducible and 
> contain X.29 and X.30 as their composition factors.
> 
> Sincerely,
> Shunsuke Tsuchioka
> 
> 
> 
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On Feb 28, 2011, at 3:32 PM, Lukas Maas wrote:

> Dear All,
> 
> this is indeed a contradiction to the paper by Morris and Yaseen. 
> But, for instance, using the MeatAxe programs it can be proved computationally 
> that there is an 1440-dimensional irreducible 3-modular spin representation of Sym(11). 
> 
> I attached a GAP-readable file that contains two matrices a and b which are preimages 
> of (1,2) and (1,..,11) under the 32-dimensional basic spin representation m32 of Sym(11) over GF(3). 
> The tensor product of m32 with itself has two composition factors of dimension 45.
> If m45 denotes one of them, then the tensor product of m45 with itself has a composition factor of dimension 131, 
> say m131, such that the wanted spin representation of dimension 1440 appears as composition factors of the tensor product of m32 with m131.
> 
> Read("mat.g");
> Order(a); #4
> Order(b); #22
> 
> # find m45 in m32 x m32
> t1:= KroneckerProduct( a, a );;
> t2:= KroneckerProduct( b, b );;
> t:= GModuleByMats( [t1,t2], GF(3) );
> cf32x32:= MTX.CompositionFactors( t );
> pos:= Position( List(cf32x32,c->c.dimension), 45 );
> m45:= cf32x32[pos].generators;
> 
> # find m131 in m45 x m45
> t1:= KroneckerProduct( m45[1], m45[1] );;
> t2:= KroneckerProduct( m45[2], m45[2] );;
> t:= GModuleByMats( [t1,t2], GF(3) );
> cf45x45:= MTX.CompositionFactors( t );
> pos:= Position( List(cf45x45,c->c.dimension), 131 );
> m131:= cf45x45[pos].generators;
> 
> # find the pair of 1440-dim. spin rep's in m32 x m131
> t1:= KroneckerProduct( a, m131[1] );;
> t2:= KroneckerProduct( b, m131[2] );;
> t:= GModuleByMats( [t1,t2], GF(3) );
> cf32x131:= MTX.CompositionFactors( t );
> List( cf32x131, c-> c.dimension );
> #[ 1440, 640, 32, 1440, 640 ]  
> 
> Cheers,
> Lukas
> 
> <mat.g>

On Feb 28, 2011, at 3:40 PM, Juergen Mueller wrote:

> Dear All,
> 
> in addition to Lukas's comment, I would just like to mention
> that the mistake in the Morris-Yaseen paper (which indeed was 
> known for a while to the Modular Atlas community) is on 
> p.163, l.-7: In the decomposition of the tensor product 
> <7,4> x [9,2] it should correctly read b = 8 + 11x and c = 8 + 10x, 
> thus the argument there just breaks down.
> 
> Best wishes, J"urgen M"uller
>