GAP Equivariant Degree Library

Author

Haopin Wu <psistwu@outlook.com>

Overview

This library provides GAP procedures for computation related to equivariant degree theory, which consists of the following parts:

1. System
2. Basic
3. Group
4. Compact Lie Group
5. Orbit Type
6. Direct Product
7. Burnside Ring

For each part listed above, there is a corresponding demo file in folder ./demo. If you need to run gap in a folder different from ./demo, please copy and paste ./demo/preload.gap to your working directory and change the path written in the file accordingly. You can then load the library in the same way as that in the demo files.

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Part 1: System

Files

• libSys.g
• libSys.gd
• libSys.gi

Summary

libSys provides some system-level functions.

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1-1 Clean (global function)

Clean()

Clean resets all user-defined variables.

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Example 1.1

gap> a := 1;;
gap> Clean();
Cleaning all user-defined variables.... Done!
gap> a;
Error, Variable: 'a' must have a value
not in any function at *stdin*:3

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Part 2: Basic

Files

• libBasic.g
• libBasic.gd
• libBasic.gi

Summary

libBasic includes basic functions for mathematics.

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2-1 DotFileLattice (operation)

DotFileLattice( lat, filename )

DotFileLattice generates a dot file with name filename associated to lattice lat. To obtain the diagram in a different format, one needs to use the following command in Bash:

dot -T<format> -o <outputfile> <inputfile>

Please see examples in Part 3: Group and Part 4: Orbit Type for concrete usage.

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Part 3: Group

Files

• libGroup.g
• libGroup.gd
• libGroup.gi

Dependency

• libBasic

Summary

libGroup provides more functions for basic group theory.

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3-1 \< (operation)

Csubg1 < Csubg2

This is the partial order on the collection of CCSs of a group induced by the subgroup relation.

3-2 OrderOfWeylGroup (attribute)

OrderOfWeylGroup( subg )
OrderOfWeylGroup( Csubg )

OrderOfWeylGroup returns the order of the Weyl group of a subgroup. The input argument can be either a subgroup or a CCS.

3-3 nLHnumber (operation)

nLHnumber( L, H )
nLHnumber( CL, CH )

nLHnumber returns $n(L,H)$, i.e., the number of subgroups conjugate to $H$ which contain $L$. The input arguments can be either two subgroups or two CCSs.

3-4 pCyclicGroup (operation)

pCyclicGroup( n )

pCyclicGroup returns a cyclic group embedded in a symmetric group.

3-5 pDihedralGroup (operation)

pDihedralGroup( n )

pDihedralGroup returns a dihedral group embedded in a symmetric group.

3-6 mCyclicGroup (operation)

mCyclicGroup( n )

mCyclicGroup returns a cyclic group as a collection of 2x2 matrices (a subgroup of $SO(2)$).

3-7 mDihedralGroup (operation)

mDihedralGroup( n )

mDihedralGroup returns a dihedral group as a collection of 2x2 matrices (a subgroup of $O(2)$).

3-8 LatticeCCSs (attribute)

LatticeCCSs( grp )

LatticeCCSs returns the CCSs lattice of group grp.

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Example 3.1

This example demonstrates how to find the order of the Weyl group and $n(L,H)$ for subgroups of $D_6$.

First, compute all CCSs of $D_6$.

gap> grp := pDihedralGroup( 6 );;
gap> ccs_list := ConjugacyClassesSubgroups( grp );;

Next, select two CCSs of $D_6$ and compute the order of the Weyl groups and $n(L,H)$.

gap> ccs1 := ccs_list[ 5 ];;
gap> ccs2 := ccs_list[ 8 ];;
gap> OrderOfWeylGroup( ccs1 );
4
gap> OrderOfWeylGroup( ccs2 );
2
gap> ccs1 < ccs2;
true
gap> nLHnumber( ccs1, ccs2 );
1

Example 3.2

This example shows how to generate CCSs lattice diagarm of $S_4$ in pdf format.

First, generate a dot file associated to the CCSs lattice diagram of $S_4$.

gap> grp := SymmetricGroup( 4 );;
gap> lat := LatticeCCSs( grp );
<CCS lattice of Sym( [ 1 .. 4 ] ), 11 classes>
gap> DotFileLattice( lat, "s4_ccslat.dot" );;

Then, convert the dot file to a pdf file by the following command in Bash.

$ dot -Tpdf -o s4_ccslat.pdf s4_ccslat.dot

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Part 4: Compact Lie Group

Files

• libCompactLieGroup.g
• libCompactLieGroup.gd
• libCompactLieGroup.gi

Dependency

• libBasic
• libGroup

Summary

libCompactLieGroup deals with Compact Lie Groups O(2) and SO(2).

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Part 5: Orbit Type

Files

• libOrbitType.g
• libOrbitType.gd
• libOrbitType.gi

Dependency

• libBasic
• libGroup

Summary

libOrbitType deals with orbit types of group representations.

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5-1 OrbitTypes (attribute)

OrbitTypes( char )

OrbitTypes returns the indices of CCSs which are orbit types of the given group representation described by its character char.

5-2 LatticeOrbitTypes (attribute)

LatticeOrbitTypes( char )

LatticeOrbitTypes returns the lattice of orbit types of the given group representation described by its character char.

5-3 DimensionOfFixedSet (operation)

DimensionOfFixedSet( char, subg )
DimensionOfFixedSet( char, Csubg )

DimensionOfFixedSet returns dimension of the fixed set of a subgroup with respect to the given group representation described by its character char. The second argement can be either a subgroup or a CCS.

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Example 5.1

This example demonstrates how to find orbit types and the lattice of an irreducible $S_4$-representation described by its character.

First, let us select a 3-dimensional irreducible $S_4$-representation.

gap> grp := SymmetricGroup( 4 );;
gap> Display( CharacterTable( grp ) );
CT2

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

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

X.1     1 -1  1  1 -1
X.2     3 -1 -1  .  1
X.3     2  .  2 -1  .
X.4     3  1 -1  . -1
X.5     1  1  1  1  1
gap> char := Irr( grp )[ 2 ];;

Next, find indices of orbit types and generate a dot file associated to the lattice.

gap> ind_orbtypes := OrbitTypes( char );
[ 1, 3, 4, 7, 11 ]
gap> lat_orbtypes := LatticeOrbitTypes( char );;
gap> DotFileLattice( lat_orbtypes, "s4_obtlat2.dot" );;

Finally, convert the dot file to a pdf file in Bash.

$ dot -Tpdf -o s4_obtlat2.pdf s4_obtlat2.dot

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Part 6: Direct Product

Files

• libDirectProduct.g
• libDirectProduct.gd
• libDirectProduct.gi

Dependency

• libBasic
• libGroup

Summary

libDirectProduct implements the amalgamation notation for subgroups of the direct product of two groups.

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6-1 DirectProductComponents (attribute)

DirectProductComponents( grp )

DirectProductComponents returns the list of direct product components of group grp.

6-2 SubgroupDirectProductInfo (attribute)

SubgroupDirectProductInfo( subg )

SubgroupDirectProductInfo returns the amalgamation description of a subgroup of the direct product of two groups.

6-3 AmalgamationQuadruple (attribute)

AmalgamationQuadruple( Csubg )

AmalgamationQuadruple returns the quadruple of indices which gives the amalgamation description of a CCS.

6-4 DirectProductDecomposition (operation)

DirectProductDecomposition( grp, elm )

DirectProductDecomposition returns the direct product decomposition of an element in the direct product of groups.

6-5 DirectProductDecomposition (attribute)

DirectProductDecomposition( csubg ) DirectProductDecomposition( char )

Suppose $G$ is a direct product of groups. DirectProductDecomposition decomposes (1) a conjugacy class of subgroups of $G$ by conjugacy classes of subgroups of the direct product components of $G$, or (2) a irreducible character of $G$ by irreducible character of direct product components of $G$.

6-6 AmalgamationNotation (operation)

AmalgamationNotation( Csubg )

AmalgamationNotation returns the amalgamation notation of a CCS of the direct product of two groups. Before calling this function, one should name all CCSs of each direct product component; otherwise, the the function will replace names by indices.

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Example 6.1

This example demonstrates how to obtain the amalgamation notation of a subgroup and a CCS of $S_4\times D_3$.

First, create the group $S_4\times D_3$ and compute its CCSs.

gap> grp1 := SymmetricGroup( 4 );;
gap> grp2 := pDihedralGroup( 3 );;
gap> grp := DirectProduct( grp1, grp2 );;
gap> ccs_list := ConjugacyClassesSubgroups( grp );;

Next, take a subgroup of $S_4\times D_3$ and the associated CCS.

gap> Csubg := ccs_list[ 38 ];;
gap> subg := representative( Csubg );;

Now, we can get the amalgamation descrpition of the subgroup.

gap> SubgroupDirectProductInfo( subg );
rec( DiscriminantElements := [ (1,2)(3,4)(6,7) ],
  Quadruple := [ Group([ (1,2)(3,4), (), (3,4) ]), Group([ (3,4) ]), Group([ (1,2,3) ]),
      Group([ (2,3), (1,2,3), () ]) ] )

In order to obtain the amalgamation notation of CCSs of $S_4\times D_3$, one has to name CCSs of $S_4$ and $D_3$.

gap> ccs_s4_list := ConjugacyClassesSubgroups( grp1 );;
gap> SetName( ccs_s4_list[ 1 ], "Z1" );;
gap> SetName( ccs_s4_list[ 2 ], "Z2" );;
gap> SetName( ccs_s4_list[ 3 ], "D1" );;
gap> SetName( ccs_s4_list[ 4 ], "Z3" );;
gap> SetName( ccs_s4_list[ 5 ], "V4" );;
gap> SetName( ccs_s4_list[ 6 ], "D2" );;
gap> SetName( ccs_s4_list[ 7 ], "Z4" );;
gap> SetName( ccs_s4_list[ 8 ], "D3" );;
gap> SetName( ccs_s4_list[ 9 ], "D4" );;
gap> SetName( ccs_s4_list[ 10 ], "A4" );;
gap> SetName( ccs_s4_list[ 11 ], "S4" );;
gap> ccs_d3_list := ConjugacyClassesSubgroups( grp2 );;
gap> SetName( ccs_d3_list[ 1 ], "Z1" );;
gap> SetName( ccs_d3_list[ 2 ], "D1" );;
gap> SetName( ccs_d3_list[ 3 ], "Z3" );;
gap> SetName( ccs_d3_list[ 4 ], "D3" );;

Then, the following function will return the amalgamation notation of the given CCS.

gap> AmalgamationNotation( c );
"(D2[D1,Z3]D3)"

Example 6.2

This example demonstrates how to decompose a CC and a irreducible representation of $S_4\times D_3$.

First, generate the group $G:=S_4\times D_3$.

gap> grp1 := SymmetricGroup( 4 );;
gap> grp2 := pDihedralGroup( 3 );;
gap> grp  := DirectProduct( grp1, grp2 );;

Then, select a conjugacy class and a irreducible character of $G$.

gap> cc_list := ConjugacyClass( grp );;
gap> irr_list := Irr( grp );;
gap> cc := cc_list[ 9 ];;
gap> irr := irr_list[ 7 ];;
gap> DirectProductDecomposition( cc );
[ (2,3,4)^G, (1,2,3)^G ]
gap> DirectProductDecomposition( irr );
[ Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, -1 ] ) ]

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Part 7: Burnside ring

Files

• libBurnsideRing.g
• libBurnsideRing.gd
• libBurnsideRing.gi

Dependency

• libBasic
• libGroup
• libOrbitType

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7-1 BurnsideRing (attribute)

BurnsideRing( grp )

BurnsideRing returns the Burnside ring induced by a group grp. Please refer to Example 6.1 for arithmetic in the Burnside ring.

7.2 Basis (attribute)

Basis( brng )

Basis returns the canonical basis of brng as a free module over integers.

7.3 UnderlyingGroup (attribute)

UnderlyingGroup( brng )

UnderlyingGroup returns the the group by which brng is induced.

7.4 Dimension (attribute)

Dimension( brng )

Dimension returns the dimension of brng as a free module over integers.

7.5 Length (attribute)

Length( e )

Length returns the length of a Burnside ring element, which is the length of the summation.

7.6 BasicDegree (attribute)

BasicDegree( char )

BasicDegree returns the basic degree of a given chararcter char.

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Example 7.1

This example demonstrates how to perform arithmetic in Burnside ring $A(S_4)$.

First, generate Burnside ring $R:=A(S_4)$.

gap> grp  := SymmetricGroup( 4 );
Sym( [ 1 .. 4 ] )
gap> brng := BurnsideRing( grp );
BurnsideRing( Sym( [ 1 .. 4 ] ) )

Let us take two elements in $R$ and perform three basic operations, i.e., addition, multiplication and scalar multiplication.

gap> a := Basis( brng )[ 4 ];
<1(4)> in BurnsideRing( Sym( [ 1 .. 4 ] ) )
gap> b := Basis( brng )[ 8 ];
<1(8)> in BurnsideRing( Sym( [ 1 .. 4 ] ) )
gap> a + b;
<1(4)+1(8)> in BurnsideRing( Sym( [ 1 .. 4 ] ) )
gap> a * b;
<1(1)+1(4)> in BurnsideRing( Sym( [ 1 .. 4 ] ) )
gap> 3 * (a + b);
<3(4)+3(8)> in BurnsideRing( Sym( [ 1 .. 4 ] ) )

Example 7.2

This example demonstrates how to calculate basic degree of an irreducible $S_4$-representation and check if the its square is multiplicative idenitity of $R$.

After performing Example 6.1, select an irreducible character.

gap> char := Irr( grp )[ 2 ];
Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 3, -1, -1, 0, 1 ] )
gap> bdeg := BasicDegree( char );
<1(1)-1(3)-1(4)-1(7)+1(11)> in BurnsideRing( Sym( [ 1 .. 4 ] ) )

Then, compute the square of the basic degree and compare with the multiplicative identity of $R$.

gap> bdeg^2;
<1(11)> in BurnsideRing( Sym( [ 1 .. 4 ] ) )
gap> One( brng );
<1(11)> in BurnsideRing( Sym( [ 1 .. 4 ] ) )
gap> bdeg^2 = One( brng );
true

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