69.1 Power Maps

The n-th power map of a character table is represented by a list that stores at position i the position of the class containing the n-th powers of the elements in the i-th class. The n-th power map can be composed from the power maps of the prime divisors p of n, so usually only power maps for primes p are actually stored in the character table.

For an ordinary character table tbl with access to its underlying group G, the p-th power map of tbl can be computed using the identification of the conjugacy classes of G with the classes of tbl. For an ordinary character table without access to a group, in general the p-th power maps (and hence also the element orders) for prime divisors p of the group order are not uniquely determined by the matrix of irreducible characters. So only necessary conditions can be checked in this case, which in general yields only a list of several possibilities for the desired power map. Character tables of the GAP character table library store all p-th power maps for prime divisors p of the group order.

Power maps of Brauer tables can be derived from the power maps of the underlying ordinary tables.

For (computing and) accessing the n-th power map of a character table, PowerMap (see PowerMap) can be used; if the n-th power map cannot be uniquely determined then PowerMap returns fail.

The list of all possible p-th power maps of a table in the sense that certain necessary conditions are satisfied can be computed with PossiblePowerMaps (see PossiblePowerMaps). This provides a default strategy, the subroutines are listed in Section Subroutines for the Construction of Power Maps.

  • PowerMap( tbl, n[, class] ) O
  • PowerMapOp( tbl, n[, class] ) O
  • ComputedPowerMaps( tbl ) A

    Called with first argument a character table tbl and second argument an integer n, PowerMap returns the n-th power map of tbl. This is a list containing at position i the position of the class of n-th powers of the elements in the i-th class of tbl.

    If the additional third argument class is present then the position of n-th powers of the class-th class is returned.

    If the n-th power map is not uniquely determined by tbl then fail is returned. This can happen only if tbl has no access to its underlying group.

    The power maps of tbl that were computed already by PowerMap are stored in tbl as value of the attribute ComputedPowerMaps, the n-th power map at position n. PowerMap checks whether the desired power map is already stored, computes it using the operation PowerMapOp if it is not yet known, and stores it. So methods for the computation of power maps can be installed for the operation PowerMapOp. 

    gap> tbl:= CharacterTable( "L3(2)" );;
gap> ComputedPowerMaps( tbl );
[ , [ 1, 1, 3, 2, 5, 6 ], [ 1, 2, 1, 4, 6, 5 ],,,, [ 1, 2, 3, 4, 1, 1 ] ]
gap> PowerMap( tbl, 5 );
[ 1, 2, 3, 4, 6, 5 ]
gap> ComputedPowerMaps( tbl );
[ , [ 1, 1, 3, 2, 5, 6 ], [ 1, 2, 1, 4, 6, 5 ],, [ 1, 2, 3, 4, 6, 5 ],, 
  [ 1, 2, 3, 4, 1, 1 ] ]
gap> PowerMap( tbl, 137, 2 );
2

  • PossiblePowerMaps( tbl, p[, options] ) O

    For the ordinary character table tbl of the group G, say, and a prime integer p, PossiblePowerMaps returns the list of all maps that have the following properties of the p-th power map of tbl. (Representative orders are used only if the OrdersClassRepresentatives value of tbl is known, see OrdersClassRepresentatives.)

    1.
    For class i, the centralizer order of the image is a multiple of the i-th centralizer order; if the elements in the i-th class have order coprime to p then the centralizer orders of class i and its image are equal.
    2.
    Let n be the order of elements in class i. If prime divides n then the images have order n/p; otherwise the images have order n. These criteria are checked in InitPowerMap (see InitPowerMap).
    3.
    For each character c of G and each element g in G, the values c(gp) and GaloisCyc( c(g), p ) are algebraic integers that are congruent modulo p; if p does not divide the element order of g then the two values are equal. This congruence is checked for the characters specified below in the discussion of the options argument; For linear characters l among these characters, the condition c(g)p = c(gp) is checked. The corresponding function is Congruences (see Congruences).
    4.
    For each character c of G, the kernel is a normal subgroup N, and gp Î N for all g Î N; moreover, if N has index p in G then gp Î N for all g Î G, and if the index of N in G is coprime to p then gp \not Î N for each g \not Î N. These conditions are checked for the kernels of all characters c specified below, the corresponding function is ConsiderKernels (see ConsiderKernels).
    5.
    If p is larger than the order m of an element g Î G then the class of gp is determined by the power maps for primes dividing the residue of p modulo m. If these power maps are stored in the ComputedPowerMaps value (see ComputedPowerMaps) of tbl then this information is used. This criterion is checked in ConsiderSmallerPowerMaps (see ConsiderSmallerPowerMaps).
    6.
    For each character c of G, the symmetrization y defined by y(g) = (c(g)p - c(gp))/p is a character. This condition is checked for the kernels of all characters c specified below, the corresponding function is PowerMapsAllowedBySymmetrizations (see PowerMapsAllowedBySymmetrizations).

    If tbl is a Brauer table, the possibilities are computed from those for the underlying ordinary table.

    The optional argument options must be a record that may have the following components:

    chars:
    a list of characters which are used for the check of the criteria 3., 4., and 6.; the default is Irr( tbl ),

    powermap:
    a parametrized map which is an approximation of the desired map

    decompose:
    a boolean; a true value indicates that all constituents of the symmetrizations of chars computed for criterion 6. lie in chars, so the symmetrizations can be decomposed into elements of chars; the default value of decompose is true if chars is not bound and Irr( tbl ) is known, otherwise false,

    quick:
    a boolean; if true then the subroutines are called with value true for the argument quick; especially, as soon as only one possibility remains this possibility is returned immediately; the default value is false,

    parameters:
    a record with components maxamb, minamb and maxlen which control the subroutine PowerMapsAllowedBySymmetrizations; it only uses characters with current indeterminateness up to maxamb, tests decomposability only for characters with current indeterminateness at least minamb, and admits a branch according to a character only if there is one with at most maxlen possible symmetrizations. 

    gap> tbl:= CharacterTable( "U4(3).4" );;
gap> PossiblePowerMaps( tbl, 2 );
[ [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, 6, 14, 9, 1, 1, 2, 2, 3, 4, 5, 6, 
      8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 20, 20, 
      20, 20, 22, 22, 24, 24, 25, 26, 28, 28, 29, 29 ] ]

  • ElementOrdersPowerMap( powermap ) F

    Let powermap be a nonempty list containing at position p, if bound, the p-th power map of a character table or group. ElementOrdersPowerMap returns a list of the same length as each entry in powermap, with entry at position i equal to the order of elements in class i if this order is uniquely determined by powermap, and equal to an unknown (see Chapter Unknowns) otherwise. 

    gap> tbl:= CharacterTable( "U4(3).4" );;
gap> known:= ComputedPowerMaps( tbl );;
gap> Length( known );
7
gap> sub:= ShallowCopy( known );;  Unbind( sub[7] );
gap> ElementOrdersPowerMap( sub );
[ 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, Unknown(1), Unknown(2), 8, 9, 12, 2, 2, 4, 4, 
  6, 6, 6, 8, 10, 12, 12, 12, Unknown(3), Unknown(4), 4, 4, 4, 4, 4, 4, 8, 8, 
  8, 8, 12, 12, 12, 12, 12, 12, 20, 20, 24, 24, Unknown(5), Unknown(6), 
  Unknown(7), Unknown(8) ]
gap> ord:= ElementOrdersPowerMap( known );
[ 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 12, 2, 2, 4, 4, 6, 6, 6, 8, 10, 
  12, 12, 12, 14, 14, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 12, 12, 12, 12, 12, 12, 
  20, 20, 24, 24, 28, 28, 28, 28 ]
gap> ord = OrdersClassRepresentatives( tbl );
true

  • PowerMapByComposition( tbl, n ) F

    tbl must be a nearly character table, and n a positive integer. If the power maps for all prime divisors of n are stored in the ComputedPowerMaps list of tbl then PowerMapByComposition returns the n-th power map of tbl. Otherwise fail is returned. 

    gap> tbl:= CharacterTable( "U4(3).4" );;  exp:= Exponent( tbl );
2520
gap> PowerMapByComposition( tbl, exp );
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
  1, 1, 1 ]
gap> Length( ComputedPowerMaps( tbl ) );
7
gap> PowerMapByComposition( tbl, 11 );
fail
gap> PowerMap( tbl, 11 );;
gap> PowerMapByComposition( tbl, 11 );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 
  22, 23, 24, 26, 25, 27, 28, 29, 31, 30, 33, 32, 35, 34, 37, 36, 39, 38, 41, 
  40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52 ]

    The permutation group of matrix automorphisms (see MatrixAutomorphisms) acts on the possible power maps returned by PossiblePowerMaps (see PossiblePowerMaps) by permuting a list via Permuted (see Permuted) and then mapping the images via OnPoints (see OnPoints). Note that by definition, the group of table automorphisms acts trivially.

  • OrbitPowerMaps( map, permgrp ) F

    returns the orbit of the power map map under the action of the permutation group permgrp via a combination of Permuted (see Permuted) and OnPoints (see OnPoints).

  • RepresentativesPowerMaps( listofmaps, permgrp ) F

    returns a list of orbit representatives of the power maps in the list listofmaps under the action of the permutation group permgrp via a combination of Permuted (see Permuted) and OnPoints (see OnPoints). 

    gap> tbl:= CharacterTable( "3.McL" );;
gap> grp:= MatrixAutomorphisms( Irr( tbl ) );  Size( grp );
<permutation group with 5 generators>
32
gap> poss:= PossiblePowerMaps( CharacterTable( "3.McL" ), 3 );
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 
      4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37, 37, 37, 40, 
      40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 
      14, 14, 37, 37, 37, 37, 37, 37 ], 
  [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 
      4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37, 37, 37, 40, 
      40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 
      14, 14, 37, 37, 37, 37, 37, 37 ] ]
gap> reps:= RepresentativesPowerMaps( poss, grp );
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 
      4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37, 37, 37, 40, 
      40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 
      14, 14, 37, 37, 37, 37, 37, 37 ] ]
gap> orb:= OrbitPowerMaps( reps[1], grp );
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 
      4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37, 37, 37, 40, 
      40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 
      14, 14, 37, 37, 37, 37, 37, 37 ], 
  [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 
      4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37, 37, 37, 40, 
      40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 14, 
      14, 14, 37, 37, 37, 37, 37, 37 ] ]
gap> Parametrized( orb );
[ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17, 4, 4, 4, 
  4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, [ 8, 9 ], [ 8, 9 ], 37, 37, 
  37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14, 14, 14, 
  14, 14, 14, 37, 37, 37, 37, 37, 37 ]

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    GAP 4 manual
    February 2000