> < ^ Date: Mon, 15 Jul 1996 15:47:30 +0300
< ^ From: Thierry Dana-Picard <dana@math.jct.ac.il >
> ^ Subject: Decomposition

Dear Forum,
While trying to compute the decomposition matrix with a modular
character table, I got a "surprising" answer:
Here is the gap session (the group is A7, the prime is 5; but I got the same
kind of answer for other primes/other groups):

```gap> a7m5:=CharTable("A7mod5");
CharTable( "A7mod5" )
gap>  DisplayCharTable(a7m5);
A7mod5
```
```2  3  3  2  .  2  2  .  .
3  2  1  2  2  .  1  .  .
5  1  .  .  .  .  .  .  .
7  1  .  .  .  .  .  1  1
```
```   1a 2a 3a 3b 4a 6a 7a 7b
2P 1a 1a 3a 3b 2a 3a 7a 7b
3P 1a 2a 1a 1a 4a 2a 7b 7a
5P 1a 2a 3a 3b 4a 6a 7b 7a
7P 1a 2a 3a 3b 4a 6a 1a 1a
```
```X.1     1  1  1  1  1  1  1  1
X.2     6  2  3  .  . -1 -1 -1
X.3     8  . -1 -1  .  3  1  1
X.4    10 -2  1  1  .  1  A /A
X.5    10 -2  1  1  .  1 /A  A
X.6    13  1 -2  1 -1 -2 -1 -1
X.7    15 -1  3  . -1 -1  1  1
X.8    35 -1 -1 -1  1 -1  .  .

A = E(7)+E(7)^2+E(7)^4
= (-1+ER(-7))/2 = b7
gap> a7m5.irreducibles;
[ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 6, 2, 3, 0, 0, -1, -1, -1 ],
[ 8, 0, -1, -1, 0, 3, 1, 1 ],
[ 10, -2, 1, 1, 0, 1, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6 ],
[ 10, -2, 1, 1, 0, 1, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4 ],
[ 13, 1, -2, 1, -1, -2, -1, -1 ], [ 15, -1, 3, 0, -1, -1, 1, 1 ],
[ 35, -1, -1, -1, 1, -1, 0, 0 ] ]
gap>  reg5:=CharTableRegular(a7,5);
CharTable( "Regular(A7,5)" )
gap>  chars5:=Restricted(a7,reg5,a7.irreducibles);
[ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 6, 2, 3, -1, 0, 0, -1, -1 ],
[ 10, -2, 1, 1, 1, 0, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4 ],
[ 10, -2, 1, 1, 1, 0, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6 ],
[ 14, 2, -1, -1, 2, 0, 0, 0 ], [ 14, 2, 2, 2, -1, 0, 0, 0 ],
[ 15, -1, 3, -1, 0, -1, 1, 1 ], [ 21, 1, -3, 1, 0, -1, 0, 0 ],
[ 35, -1, -1, -1, -1, 1, 0, 0 ] ]
gap> Decomposition(a7m5.irreducibles,chars5,"nonnegative");
[ [ 1, 0, 0, 0, 0, 0, 0, 0 ], false, false, false, false, false, false,
false, false ]
```

So what happens? And what can be done in order to get the decomposition matrix?

Thanks a lot,
Thierry.

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