> < ^ Date: Thu, 13 May 1993 12:36:57 +0200
^ From: Tom P. McDonough <tpd@aberystwyth.ac.uk >
> ^ Subject: A CharTable Problem.

The following is a brief report on some difficulties encountered in the
use of the CharTable function to get character tables of Weyl groups
of type D. A problem seems to occur for even rank. An additional
curiosity is that a character table is produced for W(D3) indicating
a group of order 24 whose identity has a centralizer of order 48.

Tom McDonough

```=========================================================================
```

Extracts from various GAP sessions, illustrating some problems
in attempting to generate the character tables for Weyl groups
of type D.

```- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
gap> CharTable("WeylD",4);
Error, Range: <low> must be an integer at
for <var>  in [ i - k + 1 .. i - 1 ] ... in
CharValueSymmetric( n, BetaSet( alpha ), pi ) called from
CharTableSymmetric.irreducibles[1][1]( n / 2, alpha[1], pi[1] / 2
) called from
gtab.irreducibles[genchar[i]][genclass[j]]( q, charparam[i], classparam[j]
) called from
fun( i ) called from
List( [ 1 .. Length( classparam ) ], function ( j ) ... end ) called from
..
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
gap> CharTable("WeylD",6);
Error, Range: <low> must be an integer at
for <var>  in [ i - k + 1 .. i - 1 ] ... in
CharValueSymmetric( n, BetaSet( alpha ), pi ) called from
CharTableSymmetric.irreducibles[1][1]( n / 2, alpha[1], pi[1] / 2
) called from
gtab.irreducibles[genchar[i]][genclass[j]]( q, charparam[i], classparam[j]
) called from
fun( i ) called from
List( [ 1 .. Length( classparam ) ], function ( j ) ... end ) called from
..
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
gap> CharTable("WeylD",8);
Error, Range: <low> must be an integer at
for <var>  in [ i - k + 1 .. i - 1 ] ... in
CharValueSymmetric( n, BetaSet( alpha ), pi ) called from
CharTableSymmetric.irreducibles[1][1]( n / 2, alpha[1], pi[1] / 2
) called from
gtab.irreducibles[genchar[i]][genclass[j]]( q, charparam[i], classparam[j]
) called from
fun( i ) called from
List( [ 1 .. Length( classparam ) ], function ( j ) ... end ) called from
..
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
These commands were successful, but I haven't checked whether the
results are correct.
gap> CharTable("WeylD",5);
gap> CharTable("WeylD",7);
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Finally, an interesting record.
gap> CharTable("WeylD",3);
rec( name := "W(D3)", order := 24, centralizers := [ 48, 8, 8, 4, 6
], orders := [ 1, 2, 2, 4, 3 ], powermap :=
[ , [ 1, 1, 1, 2, 5 ], [ 1, 2, 3, 4, 1 ] ], irreducibles :=
[ [ 3, -1, -1, 1, 0 ], [ 1, 1, -1, -1, 1 ], [ 3, -1, 1, -1, 0 ],
[ 2, 2, 0, 0, -1 ], [ 1, 1, 1, 1, 1 ] ], classparam :=
[ [ 1, [ [ 1, 1, 1 ], [  ] ] ], [ 1, [ [ 1 ], [ 1, 1 ] ] ],
[ 1, [ [ 2, 1 ], [  ] ] ], [ 1, [ [  ], [ 2, 1 ] ] ],
[ 1, [ [ 3 ], [  ] ] ] ], irredinfo := [ rec(
charparam := [ 1, [ [ 1 ], [ 1, 1 ] ] ] ), rec(
charparam := [ 1, [ [  ], [ 1, 1, 1 ] ] ] ), rec(
charparam := [ 1, [ [ 1 ], [ 2 ] ] ] ), rec(
charparam := [ 1, [ [  ], [ 2, 1 ] ] ] ), rec(
charparam := [ 1, [ [  ], [ 3 ] ] ] )
], text := "computed using generic character table for Weyl groups of type D"\
, classes := [ 1, 6, 6, 12, 8 ], operations := CharTableOps )
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
```

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