Side-note/question: if we notice bugs of a theoretical nature, should
they be posted to this forum rather than (or in addition to) the
supp...@gap-system.org address?
Incidentally, I noticed recently that the IntermediateSubgroups
routine is not working properly for some cyclic groups, like C32 and
C64, though I don't think this is due to a theoretical mistake, as I'm
sure the programmers know what the subgroup lattices look like for
these groups! :) Anyway, I submitted the bug to
supp...@gap-system.org (didn't get a response).
-William
On Fri, Mar 4, 2011 at 8:04 AM, Lukas Maas <lukas.m...@iem.uni-due.de> wrote:
> 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
>>
>>
>>
>> _______________________________________________
>> Forum mailing list
>> Forum@mail.gap-system.org
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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
>>
>
>
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>
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