> < ^ Date: Thu, 18 Feb 1993 07:03:22 +0100
> < ^ From: Jacob Hirbawi <hirbawi@commquest.com >
^ Subject: mistake in the characters of 2.S4 ?

gap-forum@samson.math.rwth-aachen.de
While we're on the subject of group characters; there seems to be
a mistake in the characters of 2.S4 . In case anyone is wondering
this all relates to John McKay's recent post on sci.math regarding
the link between the finite dicyclic groups <2,3,3>=SL(2,3),
<2,3,4>=2.S4, <2,3,5>=2.A5 and the exceptional Lie algebras E6,E7,E8.

```gap> CharTable("2.S4");
....
irreducibles :=
[ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, -1, -1, -1, 1, 1 ],
[ 2, 2, 2, 0, 0, 0, -1, -1 ], [ 3, 3, -1, 1, -1, -1, 0, 0 ],
[ 4, -4, 0, 0, 0, 0, 1, -1 ],
[ 2, -2, 0, 0, E(8)+E(8)^3, -E(8)-E(8)^3, -1, 1 ],
[ 2, -2, 0, 0, -E(8)-E(8)^3, E(8)+E(8)^3, -1, 1 ],
[ 3, 3, -1, -1, 1, 1, 0, 0 ] ],
....
```

I could not duplicate John's constructions with the above characters
but using the table below everything seems to fit nicely:

```[ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, -1, -1, -1, 1, 1 ],
[ 2, 2, 2, 0, 0, 0, -1, -1 ], [ 3, 3, -1, 1, -1, -1, 0, 0 ],
[ 4, -4, 0, 0, 0, 0, 1, -1 ],
[ 2, -2, 0, 0, E(8)^3+E(8)^5, -E(8)^3-E(8)^5, -1, 1 ],
[ 2, -2, 0, 0, -E(8)^3-E(8)^5, E(8)^3+E(8)^5, -1, 1 ],
[ 3, 3, -1, -1, 1, 1, 0, 0 ] ],
```

both tables pass the TestCharacterTable tests!

Jacob Hirbawi
JcbHrb@CERF.net

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