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 11a 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.