67.9 Other Operations for Character Tables

For an ordinary character table tbl, IsInternallyConsistent checks the consistency of the following attribute values (if stored).

-

Size, SizesCentralizers, and SizesConjugacyClasses.

-

SizesCentralizers and OrdersClassRepresentatives.

-

ComputedPowerMaps and OrdersClassRepresentatives.

-

SizesCentralizers and Irr.

-

Irr (first orthogonality relation).

For a Brauer table tbl, IsInternallyConsistent checks the consistency of the following attribute values (if stored).

-

Size, SizesCentralizers, and SizesConjugacyClasses.

-

SizesCentralizers and OrdersClassRepresentatives.

-

ComputedPowerMaps and OrdersClassRepresentatives.

-

Irr (closure under complex conjugation and Frobenius map).

If no inconsistency occurs, true is returned, otherwise each inconsistency is printed to the screen if the level of InfoWarning is at least 1 (see Info Functions), and false is returned at the end.

IsPSolvableCharacterTable for the ordinary character table tbl corresponds to IsPSolvable for the group of tbl (see IsPSolvable). p must be either a prime integer or 0.

The default method uses the attribute ComputedIsPSolvableCharacterTables for storing the computed value at position p, and calls the operation IsPSolvableCharacterTableOp for computing values that are not yet known.

gap> tbl:= CharacterTable( "Sz(8)" );;
gap> IsPSolvableCharacterTable( tbl, 2 );
false
gap> IsPSolvableCharacterTable( tbl, 3 );
true

For two ordinary character tables tbl and subtbl of a group G and its subgroup U, say, and a list fus of positive integers that describes the class fusion of U into G, IsClassFusionOfNormalSubgroup returns true if U is a normal subgroup of G, and false otherwise.

gap> tblc2:= CharacterTable( "Cyclic", 2 );;
gap> tbld8:= CharacterTable( "Dihedral", 8 );;
gap> fus:= PossibleClassFusions( tblc2, tbld8 );
[ [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ]
gap> List( fus, map -> IsClassFusionOfNormalSubgroup( tblc2, map, tbld8 ) );
[ true, false, false ]

If tbl is an ordinary character table then Indicator returns the list of n-th Frobenius-Schur indicators of the characters in the list characters; the default of characters is Irr( tbl ).

The n-th Frobenius-Schur indicator n_n(c) of an ordinary character c of the group G is given by n_n(c) = [1/(|G|)] å_{g Î G} c(gⁿ).

If tbl is a Brauer table in characteristic ¹ 2 and n = 2 then Indicator returns the second indicator.

The default method uses the attribute ComputedIndicators for storing the computed value at position n, and calls the operation IndicatorOp for computing values that are not yet known.

gap> tbl:= CharacterTable( "L3(2)" );;
gap> Indicator( tbl, 2 );
[ 1, 0, 0, 1, 1, 1 ]

returns the number and isomorphism type of polyhedral subgroups of the group with ordinary character table tbl which are generated by an element g of class c1 and an element h of class c2 with the property that the product gh lies in class c3.

According to p. 233 in NPP84, the number of polyhedral subgroups of isomorphism type V₄, D_2n, A₄, S₄, and A₅ can be derived from the class multiplication coefficient (see ClassMultiplicationCoefficient.ctbl) and the number of Galois conjugates of a class (see ClassOrbit).

The classes c1, c2 and c3 in the parameter list must be ordered according to the order of the elements in these classes.

gap> NrPolyhedralSubgroups( tbl, 2, 2, 4 );
rec( number := 21, type := "D8" )

returns the class multiplication coefficient of the classes i, j, and k of the group G with ordinary character table tbl.

The class multiplication coefficient c_i,j,k of the classes i, j, k equals the number of pairs (x,y) of elements x, y Î G such that x lies in class i, y lies in class j, and their product xy is a fixed element of class k.

In the center of the group algebra of G, these numbers are found as coefficients of the decomposition of the product of two class sums K_i and K_j into class sums,

Given the character table of a finite group G, whose classes are C₁, ..., C_r with representatives g_i Î C_i, the class multiplication coefficient c_ijk can be computed by the following formula. On the other hand the knowledge of the class multiplication coefficients admits the computation of the irreducible characters of G. (see IrrDixonSchneider).

returns the so-called class structure of the classes in the list classes, for the character table tbl of the group G. The length of classes must be at least 2.

Let C = (C₁, C₂, ..., C_n) denote the n--tuple of conjugacy classes of G that are indexed by classes. The class structure n(C) equals the number of n--tuples (g₁, g₂, ¼, g_n) of elements g_i Î C_i with g₁ g₂ ¼g_n = 1. Note the difference to the definition of the class multiplication coefficients in ClassMultiplicationCoefficient (see ClassMultiplicationCoefficient.ctbl).

n(C₁, C₂, ¼, C_n) is computed using the formula

For an ordinary character table tbl and a class position i, MatClassMultCoeffsCharTable returns the matrix [ a_ijk ]_j,k of structure constants (see ClassMultiplicationCoefficient.ctbl).

gap> tbl:= CharacterTable( "L3(2)" );;
gap> ClassMultiplicationCoefficient( tbl, 2, 2, 4 );
4
gap> ClassStructureCharTable( tbl, [ 2, 2, 4 ] );
168
gap> ClassStructureCharTable( tbl, [ 2, 2, 2, 4 ] );
1848
gap> MatClassMultCoeffsCharTable( tbl, 2 );
[ [ 0, 1, 0, 0, 0, 0 ], [ 21, 4, 3, 4, 0, 0 ], [ 0, 8, 6, 8, 7, 7 ], 
  [ 0, 8, 6, 1, 7, 7 ], [ 0, 0, 3, 4, 0, 7 ], [ 0, 0, 3, 4, 7, 0 ] ]

[Top] [Previous] [Up] [Next] [Index]