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7 Representations of point groups

7.1 Point-group matrices

Point-group matrices are representations of group elements in the point group \(G\cong\Delta/\Gamma\). These matrices can efficiently be constructed. Given any symmetry g in the point group \(G\) and a set I of all translation generators \(\gamma_i\) of the translation group \(\Gamma\), a corresponding point-group matrix can be determined through the set of the symmetry transformed elements \(g \gamma_i g^{-1}\) [CGL+23]. Point-group matrices enable the conduction of symmetry analysis on any given (supercell) model graph.

7.2 PGMatricesOfGenerators

• Signature (7.2-2): signature of the underlying triangle group

• GetTriangleGroup (7.2-3): the triangle group \(\Delta\)

• GetProperTriangleGroup (7.2-4): the proper triangle group \(\Delta^+\)

• TGQuotientName (7.2-5): name of the triangle group quotient

• GetTGQuotient (7.2-6): the triangle group quotient

• IsSparse (7.2-7): boolean, if true the point-group matrices are sparsely represented

• TGGenerators (7.2-8): list of triangle group generators a, b and c

• PGMatricesRec (7.2-9): record of three point-group matrices for the generators a, b and c of the triangle group \(\Delta\)

and is printed in the form

   PGMatricesOfGenerators(
     TriangleGroup(...), 
     ProperTriangleGroup(...),
     TGQuotient(...),
     sparse = bool,
     generators = [ a, b, c ],
     pgMatricesRec = rec( a := pgMatA, b := pgMatB, c := pgMatC )
   )

7.2-1 PGMatricesOfGenerators

Returns: the point-group matrices of the (full) triangle group generators a, b and c as a PGMatricesOfGenerators object.

Constructs the point-group matrices for the generators a, b and c of the triangle group \(\Delta\) specified by the TriangleGroup object fulltg (see 2.1), which define a representation of the point group \(G\cong\Delta/\Gamma\) specified by the triangle group quotient tgquotient, a TGQuotient object (see 2.3-1). They are constructed through the use of the embedding homomorphism from the proper triangle group \(\Delta^+\) to the triangle group \(\Delta\), where the former is specified by the ProperTriangleGroup object tg (see 2.1).

The option sparse, which takes a boolean, can be used to generate a sparse representation of the point-group matrices. If sparse is true, the point-group matrices are of the form [ [ [ rowIdx, colIdx ], entry ], ... ], where entry is the corresponding matrix element at position rowIdx and colIdx, which represent indices of the matrix rows and columns, respectively. The default is false.

7.2-2 Signature

returns the signature of the triangle group associated with pgMatGs, [ r, q, p ].

7.2-3 GetTriangleGroup

returns the triangle group on which the PGMatricesOfGenerators pgMatsGs is based as a TriangleGroup object (see 2.1).

7.2-4 GetProperTriangleGroup

returns the proper triangle group on which the PGMatricesOfGenerators pgMatsGs is based as a ProperTriangleGroup object (see 2.1).

7.2-5 TGQuotientName

returns the name of the triangle group quotient. For quotients from the library, i.e., Conder’s list, this takes the form [genus, number], number is a running index over all triangle groups for the given genus.

7.2-6 GetTGQuotient

returns the triangle group quotient on which the PGMatricesOfGenerators pgMatsGs is based as a TGQuotient object (see 2.3).

7.2-7 IsSparse

returns a boolean, true if the point-group matrices in pgMatsGs are sparse false otherwise.

7.2-8 TGGenerators

returns the triangle group generators on which the PGMatricesOfGenerators pgMatsGs is based.

7.2-9 PGMatricesRec

returns a record (see chapter Records in the GAP Reference Manual) of three point-group matrices for the triangle group generators. Each component is of the form "tgGenerator" := pgMat, where "tgGenerator" is a string "a", "b" or "c" denoting one of the generators of the triangle group and pgMat the corresponding point-group matrix.

7.2-10 EvaluatePGMatrix

returns a point-group matrix for a symmetry element symmetry in the triangle group or proper triangle group specified through fulltg or tg, respectively.

7.3 PGMatrices

• GetPGMatricesOfGenerators (7.3-2): a PGMatricesOfGenerators object for the point-group matrices corresponding of the (full) triangle group generators a, b and c

• Symmetries (7.3-3): list of symmetry elements given as words in the triangle group \(\Delta\) or proper triangle group \(\Delta^+\)

• SymmetryNames (7.3-4): list of names denoting the symmetry elements

• PGMatricesRec (7.3-5): record of point-group matrices for the symmetry elements in the triangle group \(\Delta\) or proper triangle group \(\Delta^+\)

and is printed in the form

   PGMatrices(
     PGMatricesOfGenerators( ... ), 
     symmetries = [ sym1, sym2, ... ],
     symmetryNames = [ "symName1", "symName2", ... ],
     pgMatricesRec = rec( symName1 := pgMat1, symName2 := pgMat2, ... )
   )

7.3-1 PGMatrices

Returns: list of point-group matrices as a PGMatrices object.

Constructs point-group matrices for a single symmetry element or a list of symmetry elements given as words written in terms of the rotation generators a, b, c of the triangle group \(\Delta\) or the reflection generators x, y, z of the proper triangle group \(\Delta^+\), specified by the TriangleGroup object fulltg and ProperTriangleGroup object tg, respectively (see 2.1). Each point-group matrix is constructed through a sequence of matrix multiplications of the point-group matrices in pgMatsGs, i.e., the point-group matrices of the generators a, b and c of \(\Delta\), which form representations of the point group previously specified by a corresponding triangle group quotient, fulltg and tg. The sequences are specified by the corresponding words in symmetries.

The option symNames can be used to specify the names of the symmetries. If a single symmetry is provided in the argument symmetries, symNames takes a single string, otherwise a list of strings with the same number of elements as in symmetries must be provided. The names will be used to label the corresponding point-group matrices. If symNames is not specified, the words in symmetries will used as labels instead.

If the option sparse was used in the construction of pgMatsGs, the point-group matrices will be sparsely represented as well and will be of the form [ [ [ rowIdx, colIdx ], entry ], ... ], where entry is the corresponding matrix element at position rowIdx and colIdx, which represent indices of the matrix rows and columns, respectively.

7.3-2 GetPGMatricesOfGenerators

returns the point-group matrices of the (full) triangle group generators a, b and c as a PGMatricesOfGenerators object.

7.3-3 Symmetries

returns the symmetry elements of the (proper) triangle group on which the PGMatrices pgMats is based.

7.3-4 SymmetryNames

returns the names of the symmetry elements of the (proper) triangle group on which the PGMatrices pgMats is based.

7.3-5 PGMatricesRec

returns a record (see chapter Records in the GAP Reference Manual) of point-group matrices for symmetry elements in the point group written in terms of (proper) triangle group generators. Each component is of the form SymmetryName := pgMat, where SymmetryName is a string denoting one of the symmetries used and pgMat the corresponding point-group matrix.

7.4 Export and Import

7.4-1 Exporting PGMatrices Objects

Export point-group matrices PGMatrices pgMats to the OutputTextStream output-stream or to the file at the path given by the string path. Alternatively, point-group matrices can be exported to a string.

7.4-2 Importing PGMatrices Objects

Returns: point-group matrices as a PGMatrices object (see 7.3).

Import point-group matrices from the InputTextStream input-stream or from the file at the path given by the string path. Alternatively, point-group matrices can be imported from the string string. Optionally, the triangle group and the proper triangle group can be given as a TriangleGroup object fulltg and a ProperTriangleGroup object tg, respectively, (see 2.1).

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