  
  [1X7 [33X[0;0YUtility Functions Provided by the [5XCTblLib[105X[101X[1X Package[133X[101X
  
  [33X[0;0YThis  chapter  describes  [5XGAP[105X  functions  that  are  provided by the [5XCTblLib[105X
  package but that might be of general interest.[133X
  
  [33X[0;0YFor  the  moment, there are just two features to describe, the generation of
  [5XAtlas[105X  irrationalities  from  cyclotomic integers (see Section [14X7.1[114X), and the
  generation  of  information  about  the  group structure from identifiers of
  character tables (see Section [14X7.2[114X).[133X
  
  
  [1X7.1 [33X[0;0YWrite Character Values in Terms of Atomic [5XAtlas[105X[101X[1X Irrationalities[133X[101X
  
  [1X7.1-1 CTblLib.StringOfAtlasIrrationality[101X
  
  [33X[1;0Y[29X[2XCTblLib.StringOfAtlasIrrationality[102X( [3Xcyc[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya string that describes the cyclotomic integer [3Xcyc[103X.[133X
  
  [33X[0;0YThis  function  is  intended  for expressing the cyclotomic integer [3Xcyc[103X as a
  linear  combination  of so-called [21Xatomic [5XAtlas[105X irrationalities[121X (see [CCN+85,
  p. xxvii]), with integer coefficients.[133X
  
  [33X[0;0YOften there is no [21Xoptimal[121X expression of that kind for [3Xcyc[103X, and this function
  uses certain heuristics for finding a not too bad expression. Concerning the
  character tables in the [5XAtlas[105X of Finite Groups [CCN+85], an explicit mapping
  between  the values which are computed by this function and the descriptions
  that  are  shown in the book is available, see [10XCTblLib.IrrationalityMapping[110X.
  Such  a mapping is not yet available for the character tables from the [5XAtlas[105X
  of  Brauer Characters [JLPW95], [13Xthis function is only experimental[113X for these
  tables, it is likely to be changed in the future.[133X
  
  [33X[0;0Y[2XCTblLib.StringOfAtlasIrrationality[102X is used by [2XBrowseAtlasTable[102X ([14X3.5-9[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xvalues:= List( [ "e31", "y'24+3", "r2+i", "r2+i2" ],[127X[104X
    [4X[25X>[125X [27X                  AtlasIrrationality );;[127X[104X
    [4X[25Xgap>[125X [27XList( values, CTblLib.StringOfAtlasIrrationality );[127X[104X
    [4X[28X[ "e31", "y'24+3", "z8-&3+i", "2z8" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe implementation uses the following heuristics for computing a description
  of the cyclotomic integer [3Xcyc[103X with conductor [22XN[122X, say.[133X
  
  [30X    [33X[0;6YIf  [22XN[122X is not squarefree the let [22XN_0[122X be the squarefree part of [22XN[122X, split
        [3Xcyc[103X  into  the  sum  of its odd squarefree part and its non-squarefree
        part,  and  consider  the  two  values  separately;  note that the odd
        squarefree part is well-defined by the fact that the basis of the [22XN[122X-th
        cyclotomic  field  given  by  [2XZumbroichBase[102X ([14XReference: ZumbroichBase[114X)
        contains  all  primitive  [22XN_0[122X-th roots of unity. Also note that except
        for  quadratic  irrationalities  (where [22XN[122X is squarefree), all roots of
        unity   that   are   involved   in   the   representation   of  atomic
        irrationalities  w. r. t. this  basis  have  the  same  multiplicative
        order.[133X
  
  [30X    [33X[0;6YIf  [3Xcyc[103X is a multiple of a root of unity then write it as such, i. e.,
        as a string involving [22Xz_N[122X.[133X
  
  [30X    [33X[0;6YOtherwise,  if [3Xcyc[103X lies in a quadratic number field then write it as a
        linear  combination  of  an  integer. Usually the string involves [22Xr_N[122X,
        [22Xi_N[122X,  or  [22Xb_N[122X,  but also multiples of [22Xb_M[122X may occur, where [22XM[122X is a –not
        squarefree– multiple of [22XN[122X.[133X
  
  [30X    [33X[0;6YOtherwise,  find  a  large  cyclic  subgroup  of the stabilizer of [3Xcyc[103X
        inside  the  Galois group over the Rationals –this subgroup defines an
        atomic  irrationality–  and express [3Xcyc[103X as a linear combination of the
        orbit  sums. In the worst case, there is no nontrivial stabilizer, and
        we find only a description as a sum of roots of unity.[133X
  
  [33X[0;0YThere  is  of course a lot of space for improvements. For example, one could
  use  the  Bosma  basis  representation  (see  [2XBosmaBase[102X  ([14X6.5-1[114X)) of [3Xcyc[103X for
  splitting  the  value  into a sum of values from strictly smaller cyclotomic
  fields, which would be useful at least if their conductors are coprime. Note
  that  the  Bosma basis of the [22XN[122X-th cyclotomic field has the property that it
  is  a  union  of  bases for the cyclotomic fields with conductor dividing [22XN[122X.
  Thus  one can easily find out that [22Xsqrt{5} + sqrt{7}[122X can be written as a sum
  of  two  values  in  terms  of  [22X5[122X-th and [22X7[122X-th roots of unity. In non-coprime
  situations,  this  argument  fails.  For  example, one can still detect that
  [22Xsqrt{15}  + sqrt{21}[122X involves only [22X15[122X-th and [22X21[122X-th roots of unity, but it is
  not obvious how to split the value into the two parts.[133X
  
  
  [1X7.2 [33X[0;0YCreate a String that Describes the Group Structure[133X[101X
  
  [1X7.2-1 StructureDescriptionCharacterTableName[101X
  
  [33X[1;0Y[29X[2XStructureDescriptionCharacterTableName[102X( [3Xname[103X ) [32X function[133X
  
  [33X[0;0YFor  a  string  [3Xname[103X  that  is  an  admissible  name  of  a character table,
  [2XStructureDescriptionCharacterTableName[102X  returns a string that is intended as
  a description of the structure of the underlying group.[133X
  
  [33X[0;0YNote  that  many identifiers of character tables (see [2XIdentifier[102X ([14XReference:
  Identifier  for character tables[114X)) do not describe the group structure in an
  appropriate way. One reason for choosing such identifiers on purpose is that
  several  character  tables  for  isomorphic  groups  can be contained in the
  library, because the groups have different class fusions into another group.
  For  example,  the  Mathieu  group  [22XM_12[122X  contains  two classes of subgroups
  isomorphic   with   [22XM_11[122X,  and  the  identifiers  of  the  character  tables
  corresponding to these subgroups are [10X"M11"[110X and [10X"M12M2"[110X, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XStructureDescriptionCharacterTableName( "M12M2" );[127X[104X
    [4X[28X"M11"[128X[104X
  [4X[32X[104X
  
