68.13 Possible Permutation Characters

For groups H and G with H £ G, the induced character (1G)H is called the permutation character of the operation of G on the right cosets of H. If only the character table of G is available and not the group G itself, one can try to get information about possible subgroups of G by inspection of those G-class functions that might be permutation characters, using that such a class function p must have at least the following properties. (For details, see Isa76, Theorem 5.18.)

(a)
p is a character of G,
(b)
p(g) is a nonnegative integer for all g Î G,
(c)
p(1) divides |G|,
(d)
p(gn) ³ p(g) for g Î G and integers n,
(e)
[p,1G] = 1,
(f)
the multiplicity of any rational irreducible G-character y as a constituent of p is at most y(1)/[y,y],
(g)
p(g) = 0 if the order of g does not divide |G|/p(1),
(h)
p(1) |NG(g)| divides p(g) |G| for all g Î G,
(i)
p(g) £ (|G| - p(1)) / (|gG| |GalG(g)|) for all nonidentity g Î G, where |GalG(g)| denotes the number of conjugacy classes of G that contain generators of the group ág ñ,
(j)
if p is a prime that divides |G|/p(1) only once then s/(p-1) divides |G|/p(1) and is congruent to 1 modulo p, where s is the number of elements of order p in the (hypothetical) subgroup H for which p = (1H)G holds. (Note that s/(p-1) equals the number of Sylow p subgroups in H.)
Any G-class function with these properties is called a possible permutation character in GAP.

(Condition (d) is checked only for those power maps that are stored in the character table of G; clearly (d) holds for all integers if it holds for all prime divisors of the group order |G|.)

GAP provides some algorithms to compute possible permutation characters (see PermChars), and also provides functions to check a few more criteria whether a given character can be a transitive permutation character (see TestPerm1).

Some information about the subgroup U can be computed from the permutation character (1U)G using PermCharInfo (see PermCharInfo).

  • PermCharInfo( tbl, permchars ) F

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

    contained:
    a list containing, for each character y = (1U)G in permchars, a list containing at position i the number y[i] |U| / SizesCentralizers( tbl )[i], which equals the number of those elements of U that are contained in class i of tbl,

    bound:
    a list containing, for each character y = (1U)G in permchars, a list containing at position i the number |U| / gcd( |U|, SizesCentralizers( tbl )[i] ), which divides the class length in U of an element in class i of tbl,

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

    ATLAS:
    a list of strings describing the decomposition of the permutation characters in permchars into the irreducible characters of tbl, given in an ATLAS-like notation. This means that the irreducible constituents are indicated by their degrees followed by lower case letters a, b, c, ¼, which indicate the successive irreducible characters of tbl of that degree, in the order in which they appear in Irr( tbl ). A sequence of small letters (not necessarily distinct) after a single number indicates a sum of irreducible constituents all of the same degree, an exponent of the form lett^n means the letter lett is repeated n times. 

    gap> t:= CharacterTable( "A6" );;
gap> psi:= Sum( Irr( t ){ [ 1, 3, 6 ] } );
Character( CharacterTable( "A6" ), [ 15, 3, 0, 3, 1, 0, 0 ] )
gap> info:= PermCharInfo( t, psi );
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> Display( t, info.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
gap> j1:= CharacterTable( "J1" );;
gap> psi:= TrivialCharacter( CharacterTable( "7:6" ) )^j1;
Character( CharacterTable( "J1" ), [ 4180, 20, 10, 0, 0, 2, 1, 0, 0, 0, 0, 0, 
  0, 0, 0 ] )
gap> PermCharInfo( j1, psi ).ATLAS;
[ "1a+56aabb+76aaab+77aabbcc+120aaabbbccc+133a^{4}bbcc+209a^{5}" ]

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    GAP 4 manual
    February 2000