^ From:

> ^ Subject:

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

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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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