GAP (repsn) - Chapter 2: Irreducible Representations

Let G be a finite group and χ be an ordinary irreducible character of G. In this chapter we introduce some functions to construct a complex representation R of G affording χ. We proceed recursively, reducing the problem to smaller subgroups of G or characters of smaller degree until we obtain a problem which we can deal with directly. Inputs of most of the functions are a given group G, and an irreducible character χ. The output is a mapping (representation) which assigns to each generator x of G a matrix R(x). We can use these functions for all groups and all irreducible characters χ of degree less than 100 although in principle the same methods can be extended to characters of larger degree. The main methods in these functions which are used to construct representations of finite groups are Induction, Extension, Tensor Product and Dixon's method (for constructing representations of simple groups and their covers) [DA05], and Projective Representation method [DD10].

2.1 Constructing Representations

This section introduces the main function to compute a representation of a finite group G affording an irreducible character χ of G.

2.1-1 IrreducibleAffordingRepresentation

called with an irreducible character chi of a group G, this function returns a mapping (representation) which maps each generator of G to a d*d matrix, where d is the degree of chi. The group generated by these matrices (the image of the map) is a matrix group which is isomorphic to G modulo the kernel of the map. If G is a solvable group then there is no restriction on the degree of chi. In the case that G is not solvable and the character chi has degree bigger than 100 the output maybe is not correct. In this case sometimes the output mapping does not afford the given character or it does not return any mapping.

gap> s := PerfectGroup( 129024, 2 );;
gap> G := Image(IsomorphismPermGroup( s ));;
gap> chi := Irr( G )[36];; 
gap> chi[1];
64
gap> IrreducibleAffordingRepresentation( chi );; 
#I  Warning: EpimorphismSchurCover via Holt's algorithm is under construction
gap> time; 
92657

2.1-2 IsAffordingRepresentation

If chi and rep are a character and a representation of a group G, respectively, then IsAffordingRepresentation returns true if the trace of rep(x) equals chi(x) for all elements x in G.

gap> G := GL(2,7);;
gap> chi := Irr(G)[ 29 ];;
gap> rep := IrreducibleAffordingRepresentation( chi );
CompositionMapping( [(8,15,22,29,36,43)(9,16,23,30,37,44)
(10,17,24,31,38,45)(11,18,25,32,39,46)(12,19,26,33,40,47)
(13,20,27,34,41,48)(14,21,28,35,42,49), (2,29,12)(3,36,20)
(4,43,28)(5,8,30)(6,15,38)(7,22,46)(9,44,14)(10,16,17)
(11,37,27)(13,23,39)(18,24,25)(19,45,35)(21,31,47)
(26,32,33)(34,40,41)(42,48,49) ] ->
[ [ [ 0, 0, 0, -1, 0, 0, 0 ],
    [ 1, 0, -1, -1, 1, 0, -1 ] 
    [ 2, -1, -2, -2, 1, 2, -1 ],
    [ 0, 0, -1, 0, 0, 0, 0 ],
    [ 1, 0, -2, 0, 0, 1, -1 ],
    [ 1, 0, -2, -1, 1, 1, -1 ],
    [ -2, 1, 1, 1, -1, -1, 0 ] ],
  [ [ 1, -1, -1, -1, 0, 2, -1 ],
    [ 0, 0, 1, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 1, 0 ],
    [ 0, 1, -1, 0, 0, 0, -1 ],
    [ 0, 1, 0, 1, 0, -1, 0 ],
    [ 0, 1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, -1, 0, 0 ] ] ], (action isomorphism) )
gap> IsAffordingRepresentation( chi, rep );
true

We can obtain the size of the image of this representation by Size(Image(rep)) and compute the value for an arbitrary element x in G by x^rep.

2.2 Induction

2.2-1 InducedSubgroupRepresentation

computes a representation of G induced from the representation rep of a subgroup H of G. If rep has degree d then the degree of the output representation is d*|G:H|.

gap> G := SymmetricGroup( 6 );;
gap> H := AlternatingGroup( 6 );;
gap> chi := Irr( H )[ 2 ];;
gap> rep := IrreducibleAffordingRepresentation( chi );;
gap> InducedSubgroupRepresentation( G, rep ); 
[ (1,2,3,4,5,6), (1,2) ] ->
[ [ [ 0, 0, 0, 0, 0, 1, 1, -1, -1, -1 ],
    [ 0, 0, 0, 0, 0, 1, 0, -1, 0, -1 ],
    [ 0, 0, 0, 0, 0, 1, 0, 0, -1, -1 ],
    [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 1, -1, 0, -1 ],
    [ 1, 1, -1, -1, -1, 0, 0, 0, 0, 0 ],
    [ 1, 0, 0, -1, -1, 0, 0, 0, 0, 0 ],
    [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 1, 0, -1, 0, -1, 0, 0, 0, 0, 0 ],
    [ 0, 1, 0, -1, -1, 0, 0, 0, 0, 0 ] ],
  [ [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
    [ 0, 0, 0, 0, 0, 1, 1, -1, -1, -1 ],
    [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
    [ 1, 1, -1, -1, -1, 0, 0, 0, 0, 0 ] ] ]

2.3 Extension

In this section we introduce some functions for extending a representation of a subgroup to the whole group.

2.3-1 ExtendedRepresentation

Suppose H is a subgroup of a group G and chi is an irreducible character of G such that the restriction of chi to H, phi say, is irreducible. If rep is an irreducible representation of H affording phi then ExtendedRepresentation extends the representation rep of H to a representation of G affording chi. This function call can be quite expensive when the representation rep has a large degree.

	
gap> G := AlternatingGroup( 6 );;
gap> H := Group([ (1,2,3,4,6), (1,4)(5,6) ]);;
gap> chi := Irr( G )[ 2 ];;
gap> phi := RestrictedClassFunction( chi, H );;
gap> IsIrreducibleCharacter( phi );
true
gap> rep := IrreducibleAffordingRepresentation( phi );;
gap> ext := ExtendedRepresentation( chi, rep );
#I  Need to extend a representation of degree 5. This may take a while.
[ (1,2,3,4,5), (4,5,6) ] -> [
[ [ 0, 1, 0, -1, -1 ],
  [ 0, 0, 0, 1, 0 ],
  [ -1, -1, -1, 0, 0 ],
  [ 0, 0, 0, 0, -1 ],
  [ 0, 0, 1, 1, 1 ] ],
[ [ 1, 0, 1, 0, 1 ],
  [ 0, 1, 0, 0, 0 ],
  [ -1, -1, 0, 1, 0 ],
  [ 1, 1, 1, 0, 0 ],
  [ 0, 0, -1, 0, 0 ] ] ]
gap> IsAffordingRepresentation( chi, ext );
true

2.3-2 ExtendedRepresentationNormal

Suppose H is a normal subgroup of a group G and chi is an irreducible character of G such that the restriction of chi to H, phi say, is irreducible. If rep is an irreducible representation of H affording phi then ExtendedRepresentationNormal extends the representation rep of H to a representation of G affording chi. This function is more efficient than ExtendedRepresentation.

gap> G := GL(2,7);;
gap> chi := Irr( G )[ 29 ];;
gap> H := SL(2,7);;
gap> phi := RestrictedClassFunction( chi, H );;
gap> IsIrreducibleCharacter( phi );
true
gap> rep := IrreducibleAffordingRepresentation( phi );;
gap> ext := ExtendedRepresentationNormal( chi, rep );
#I  Need to extend a representation of degree 7. This may take a while.
CompositionMapping( [(8,15,22,29,36,43)(9,16,23,30,37,44)
 (10,17,24,31,38,45)(11,18,25,32,39,46)(12,19,26,33,40,47)
 (13,20,27,34,41,48)(14,21,28,35,42,49),(2,29,12)(3,36,20)
 (4,43,28)(5,8,30)(6,15,38)(7,22,46)(9,44,14)(10,16,17)
 (11,37,27)(13,23,39)(18,24,25)(19,45,35)(21,31,47)
 (26,32,33)(34,40,41)(42,48,49) ] ->
[ [ [ -1, 0, 0, 1, 0, -1, 0 ], [ -1, 0, 0, 0, 0, 0, 0 ],
  [ -1, 1, 0, 0, -1, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0 ],
  [ -1, -1, 1, 0, 1, -1, 0 ], [ 0, 0, 0, -1, 0, 0, 0 ],
  [ -1, 0, 1, -1, 1, 0, -1 ] ],
  [ [ 1, -1, 0, 1, 0, -1, 1 ], [ 1, 0, -1, 1, -1, 0, 1 ],
  [ 1, -1, 0, 1, -1, 0, 1 ], [ 0, 0, -1, 0, 0, 0, 0 ],
  [ -1, 0, 0, 1, 0, -1, 0 ], [ -1, 0, 0, 0, 0, 0, 0 ],
  [ -1, 1, 0, 0, -1, 0, 0 ] ] ], (action isomorphism) )	
gap> IsAffordingRepresentation( chi, ext );
true

2.4 Character Subgroups

If χ is an irreducible character of a group G and H is a subgroup of G such that the restriction of χ to H has a linear constituent with multiplicity one, then we call H a character subgroup relative to χ or a χ-subgroup.

2.4-1 CharacterSubgroupRepresentation

returns a representation affording chi by finding a chi-subgroup and using the method described in [Dix93]. If the second argument is a chi-subgroup then it returns a representation affording chi without searching for a chi-subgroup. In this case an error is signalled if no chi-subgroup exists.

2.4-2 IsCharacterSubgroup

	
gap> G := AlternatingGroup( 8 );;
gap> chi := Irr( G )[ 2 ];;
gap> H := AlternatingGroup( 3 );;
gap> IsCharacterSubgroup( chi, H );
true
gap> rep := CharacterSubgroupRepresentation( chi, H );
[ (1,2,3,4,5,6,7), (6,7,8) ] -> [ [ [
 1/3*E(3)+2/3*E(3)^2, 0, 0, -E(3), 0, -1/3*E(3)-2/3*E(3)^2, 1 ],
   [ 2/3*E(3)+4/3*E(3)^2, 0, 1, 0, 0, 1/3*E(3)-1/3*E(3)^2, 0 ],
   [ 2/3*E(3)+4/3*E(3)^2, 0, 0, 1, 0, 1/3*E(3)-1/3*E(3)^2, 0 ],
   [ E(3)^2, 0, 0, 0, 0, 0, 0 ],
   [ 2/3*E(3)+4/3*E(3)^2, 0, 0, 0, 1, 1/3*E(3)-1/3*E(3)^2, 0 ],
   [ -2/3*E(3)-1/3*E(3)^2, 0, 0, -1, 0, 2/3*E(3)+1/3*E(3)^2, E(3)^2 ],
   [ 0, 1, 0, 0, 0, 0, 0 ] ],
 [ [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0 ],
   [ 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0 ],
   [ 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0 ],
   [ 0, 0, 0, -E(3), E(3), 0, 1 ] ] ]

2.4-3 AllCharacterPSubgroups

returns a list of all p-subgroups of G which are chi-subgroups. The subgroups are chosen up to conjugacy in G.

2.4-4 AllCharacterStandardSubgroups

returns a list containing well described subgroups of G which are chi-subgroups. This list may contain Sylow subgroups and their derived subgroups, normalizers and centralizers in G.

2.4-5 AllCharacterSubgroups

returns a list of all chi-subgroups of G among the lattice of subgroups. This function call can be quite expensive for larger groups. The call is expensive in particular if the lattice of subgroups of the given group is not yet known.

2.5 Equivalent Representation

2.5-1 EquivalentRepresentation

computes an equivalent representation to an irreducible representation rep by transforming rep to a new basis by spinning up one vector (i.e. getting the other basis vectors as images under the first one under words in the generators). If the input representation, rep, is reducible then EquivalentRepresentation does not return any mapping. In this case see section 3.

gap> G := SymmetricGroup( 7 );;
gap> chi := Irr( G )[ 2 ];;
gap> rep := CharacterSubgroupRepresentation( chi );;
gap> equ := EquivalentRepresentation( rep );
[ (1,2,3,4,5,6,7), (1,2) ] ->
[ [ [ 0, 0, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1, -E(5)-E(5)^2-E(5)^3-2*E(5)^4 ],
   [ E(5)^3-E(5)^4, E(5)^2+E(5)^3+E(5)^4, E(5)+E(5)^3-E(5)^4, -E(5)+E(5)^2
          -3*E(5)^3-E(5)^4, -E(5)-E(5)^3+E(5)^4, 2*E(5)-2*E(5)^2+2*E(5)^3 ]
    , [ 0, 0, 0, 1, 0, 0 ],
   [ 0, 4/5*E(5)+3/5*E(5)^2+2/5*E(5)^3+1/5*E(5)^4, E(5), 1, -E(5),
       6/5*E(5)+2/5*E(5)^2+3/5*E(5)^3+4/5*E(5)^4 ], [ 0, 1, 0, 0, 0, 0 ],
   [ 0, 0, E(5), 1, -E(5), 2*E(5)+E(5)^2+E(5)^3+E(5)^4 ] ],
 [ [ -1, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -E(5)-E(5)^2-3*E(5)^4,
    -E(5)-E(5)^2-E(5)^3-2*E(5)^4, E(5)+E(5)^2+3*E(5)^4 ],
  [ 0, -1, 0, 0, 0, 0 ],
  [ 0, 0, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1, -E(5)-E(5)^2-E(5)^3-2*E(5)^4
     ], [ 0, 0, -1, -E(5)^4, 1, E(5)+E(5)^2+E(5)^3+2*E(5)^4 ],
  [ 0, 0, -E(5)^4, -E(5)^3+E(5)^4, E(5)+E(5)^2+E(5)^3+2*E(5)^4,
      E(5)^3-E(5)^4 ], [ 0, 0, 0, 0, 0, -1 ] ] ]
gap> IsAffordingRepresentation( chi, equ );
true

2 Irreducible Representations