50.28 PermCharInfo

PermCharInfo( tbl, permchars )

Let tbl be the character table of the group G, and permchars the permutation character (1_U)^G for a subgroup U of G, or a list of such characters. PermCharInfo returns a record with components

contained:

a list containing for each character in permchars a list containing at position i the number of elements of U that are contained in class i of tbl, this is equal to 'permchar[<i>]' |U| / 'tbl.centralizers[<i>]',

bound:

Let permchars[k] be the permutation character (1_U)^G. Then the class length in U of an element in class i of tbl must be a multiple of the value 'bound[<k>][<i>]' = |U| / gcd( |U|, '<tbl>.centralizers[<i>]' ),

display:

a record that can be used as second argument of DisplayCharTable to display the permutation characters and the corresponding components contained and bound, for the classes where at least one character of permchars is nonzero,

ATLAS:

list of strings containing for each character in permchars the decomposition into tbl.irreducibles in ATLAS notation.

    gap> t:= CharTable("A6");;
    gap> PermCharInfo( t, [ 15, 3, 0, 3, 1, 0, 0 ] );
    rec(
      contained := [ [ 1, 9, 0, 8, 6, 0, 0 ] ],
      bound := [ [ 1, 3, 8, 8, 6, 24, 24 ] ],
      display := rec(
          classes := [ 1, 2, 4, 5 ],
          chars := [ [ 15, 3, 0, 3, 1, 0, 0 ], [ 1, 9, 0, 8, 6, 0, 0 ],
              [ 1, 3, 8, 8, 6, 24, 24 ] ],
          letter := "I" ),
      ATLAS := [ "1a+5b+9a" ] )
    gap> DisplayCharTable( t, last.display );
    A6

         2  3  3  .  2
         3  2  .  2  .
         5  1  .  .  .

           1a 2a 3b 4a
        2P 1a 1a 3b 2a
        3P 1a 2a 1a 4a
        5P 1a 2a 3b 4a

    I.1    15  3  3  1
    I.2     1  9  8  6
    I.3     1  3  8  6

Previous Up Top Next
Index