The following file formats are introduced by the HyperCells package:
HyperCell cell graph (*.hcc) for storing the cell graphs, i.e., TGCellGraph objects (see 3.7)
HyperCell model graph (*.hcm) for storing the model graphs, i.e., TGCellModelGraph objects (see 4.1)
HyperCell supercell graph (*.hcs) for storing the supercell model graphs, i.e., TGSuperCellModelGraph objects (see 5.1)
These file formats are described in detail below.
The basic file structure of a HyperCell cell graph file is as follows (with placeholders marked by <...>):
HyperCells HCC version 1.0 [ <r>, <q>, <p> ] [ <relators> , <center> ] [ g1, g2, ... ] -> <generators> <TDGAMMA> [ <TGGx>, <TGGy>, <TGGz> ] <vertices> <edges> <edge translations> <faces> <boundary>
The first line specifies the file format and its version.
The second line gives the signature (r,q,p) of the underlying triangle group \Delta(r,q,p) and the third line specifies the triangle-group quotient \Delta^+(r,q,p)/\Gamma by the list of relators <relators> given as elements of the proper triangle group \Delta^+ with generators x, y, z and the center of the cell given as the integer <center> with 1, 2, 3 corresponding to the center lying at the vertex x, y, z of the fundamental Schwarz triangle, respectively.
The fourth line specifies the generators of the translation group \Gamma. To the left of -> are the generators of the finitely presented group and to the right the list <generators> of their images in the proper triangle group \Delta^+ in terms of the generators x, y, z.
The fifth line gives the transversal T_{\Delta^+}(\Gamma) as a list <TDGAMMA> of elements of the proper triangle group \Delta^+ and the sixth line specifies the transversals T_{G^+}(G_w^+) as lists <TGGw> with w=x,y,z (w=x,y,z) of preimages of elements of the quotient group G^+\cong\Delta^+/\Gamma under the quotient homomorphism \Delta^+\to G^+, i.e., in terms of the generators x, y, z.
Lines seven to ten specify the vertices, edges, edge translations and faces of the cell:
The vertices are given as a list <vertices> with entries of the form [ w, gi ], where w is 1, 2, 3 corresponding to the vertex x, y, z of the fundamental Schwarz triangle and gi is the position of the element in the transversal T_{G^+}(G_w^+).
The edges are given as a list <edges> with entries of the form [ v1, v2, s ], where v1 and v2 are the positions of the vertices in <vertices> connected by the edge, s is the position of the element in T_{\Delta^+}(\Gamma) labeling the Schwarz triangle this edge is a part of.
The edge translations, i.e., the translations relating the unit cell containing the first vertex (v1) to the unit cell containing the second (v2) are given as a list <edge translations> of elements of the finitely presented group, i.e., in terms of the generators given to the left of -> in the third line. If v1 corresponds to the vertex (w, [g_{u_w}]_w) and v2 to the vertex (w', [g_{\tilde{u}_{w'}}]_{w'}), then gam corresponds to the translation \gamma such that
g_{\tilde{u}_{w'}} = g_{u_w}\gamma.
The faces are given as a list <faces> with entries of the form [ edge, orientation ], where edge is the position of the edge in <edges> and orientation is 1 or -1 depending on whether the edge is traversed in the same or opposite direction as the edge given in <edges>.
Finally, the eleventh line specifies the boundary identifications of the cell as a list <boundary> with entries of the form [ d1, d2, e, b, m, gam ], where d1, d2 are elements of T_{\Delta^+}(\Gamma)\subset\Delta^+ labeling the Schwarz triangles this edge is a part of, e the position in <edges>, b a running integer index of the boundary (identical for identified boundaries), m either 0 or 1 indicating the orientation of the triangle associated with this instance of the boundary (0 for gray and 1 for white Schwarz triangle), and gam the translation relating the given to the identified boundary, as an element of the finitely presented group, i.e., in terms of the generators given to the left of -> in the third line.
The full file structure of a HyperCell cell graph file thus is as follows:
HyperCells HCC version 1.0 [ <r>, <q>, <p> ] [ [ <rel1>, <rel2>, ... ], <center> ] [ g1, g2, ... ] -> [ <generator1>, <generator2>, ... ] [ <d1>, <d2>, ... ] [ [ <gx1>, <gx2>, ... ], [ <gy1>, <gy2>, ... ], [ <gz1>, <gz2>, ... ] ] [ [ w, gi ], ... ] [ [ v1, v2, s ], ... ] [ [ gam1, gam2, ... ], ... ] [ [ edge, orientation ], ... ] [ [ d1, d2, e, b, m, gam ], ... ]
A complete example for the cell graph of the quotient T2.6 of \Delta(2,8,8) is given below:
HyperCells HCC version 1.0 [ 2, 8, 8 ] [ [ x^2, x*y*z, x*z*y, y^3*z^-1 ], 3 ] [ g1, g2, g3, g4 ] -> [ z^-1*x^-1*y*z^-1*y*z*y*x^-1, z^-1*x^-1*y*z^-1*y*z*y* x^-1*y*x*z, z*y*(x^-1*z^-1)^2*y^3*x*z, y^-1*z*y^-1*x*z ] [ 1, y^-1*x^-1, (y^-1*x^-1)^2, (y^-1*x^-1)^3, (y^-1*x^-1)^4, (y^-1*x^-1)^5, (y^-1*x^-1)^6, (y^-1*x^-1)^7 ] [ [ 1, (y^-1*x^-1)^3, y^-1*x^-1, (y^-1*x^-1)^2 ], [ 1 ], [ 1 ] ] [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 3, 1 ] ] [ [ 1, 6, 1 ], [ 5, 1, 1 ], [ 6, 5, 1 ], [ 3, 6, 2 ], [ 5, 3, 2 ], [ 6, 5, 2 ], [ 4, 6, 3 ], [ 5, 4, 3 ], [ 6, 5, 3 ], [ 2, 6, 4 ], [ 5, 2, 4 ], [ 6, 5, 4 ], [ 1, 6, 5 ], [ 5, 1, 5 ], [ 6, 5, 5 ], [ 3, 6, 6 ], [ 5, 3, 6 ], [ 6, 5, 6 ], [ 4, 6, 7 ], [ 5, 4, 7 ], [ 6, 5, 7 ], [ 2, 6, 8 ], [ 5, 2, 8 ], [ 6, 5, 8 ] ] [ 1, 1, 1, 1, g2^-1*g3*g4^-1, g4*g3^-1*g2, 1, g2^-1*g1, g1^-1*g2, 1, g4^-1, g4, g1, g2^-1*g3*g4^-1, g1^-1*g2*g3^-1*g4, g2, g2^-1*g1, g1^-1, g3, g4^-1, g4*g3^-1, g4, g4^-1*g3*g2^-1*g1, g1^-1*g2*g3^-1 ] [ [ [ [ 1, 1 ], [ 3, 1 ], [ 18, -1 ], [ 13, -1 ] ], [ [ 4, 1 ], [ 6, 1 ], [ 21, -1 ], [ 16, -1 ] ], [ [ 7, 1 ], [ 9, 1 ], [ 24, -1 ], [ 19, -1 ] ], [ [ 10, 1 ], [ 12, 1 ], [ 3, -1 ], [ 22, -1 ] ], [ [ 13, 1 ], [ 15, 1 ], [ 6, -1 ], [ 1, -1 ] ], [ [ 16, 1 ], [ 18, 1 ], [ 9, -1 ], [ 4, -1 ] ], [ [ 19, 1 ], [ 21, 1 ], [ 12, -1 ], [ 7, -1 ] ], [ [ 22, 1 ], [ 24, 1 ], [ 15, -1 ], [ 10, -1 ] ] ], [ [ [ 2, 1 ], [ 1, 1 ], [ 22, -1 ], [ 11, -1 ] ], [ [ 5, 1 ], [ 4, 1 ], [ 1, -1 ], [ 14, -1 ] ], [ [ 8, 1 ], [ 7, 1 ], [ 4, -1 ], [ 17, -1 ] ], [ [ 11, 1 ], [ 10, 1 ], [ 7, -1 ], [ 20, -1 ] ], [ [ 14, 1 ], [ 13, 1 ], [ 10, -1 ], [ 23, -1 ] ], [ [ 17, 1 ], [ 16, 1 ], [ 13, -1 ], [ 2, -1 ] ], [ [ 20, 1 ], [ 19, 1 ], [ 16, -1 ], [ 5, -1 ] ], [ [ 23, 1 ], [ 22, 1 ], [ 19, -1 ], [ 8, -1 ] ] ], [ [ [ 3, 1 ], [ 2, 1 ], [ 14, -1 ], [ 6, -1 ] ], [ [ 6, 1 ], [ 5, 1 ], [ 17, -1 ], [ 9, -1 ] ], [ [ 9, 1 ], [ 8, 1 ], [ 20, -1 ], [ 12, -1 ] ], [ [ 12, 1 ], [ 11, 1 ], [ 23, -1 ], [ 15, -1 ] ], [ [ 15, 1 ], [ 14, 1 ], [ 2, -1 ], [ 18, -1 ] ], [ [ 18, 1 ], [ 17, 1 ], [ 5, -1 ], [ 21, -1 ] ], [ [ 21, 1 ], [ 20, 1 ], [ 8, -1 ], [ 24, -1 ] ], [ [ 24, 1 ], [ 23, 1 ], [ 11, -1 ], [ 3, -1 ] ] ] ] [ [ 1, 1, 2, 1, 0, g1 ], [ (y*x)^3*y, (y*x)^3*y, 2, 1, 1, g1^-1 ], [ y^-1*x^-1, y^-1*x^-1, 5, 2, 0, g2 ], [ (y*x)^2*y, (y*x)^2*y, 5, 2, 1, g2^-1 ], [ (y^-1*x^-1)^2, (y^-1*x^-1)^2, 8, 3, 0, g3 ], [ y*x*y, y*x*y, 8, 3, 1, g3^-1 ], [ (y^-1*x^-1)^3, (y^-1*x^-1)^3, 11, 4, 0, g4 ], [ y, y, 11, 4, 1, g4^-1 ], [ (y^-1*x^-1)^4, (y^-1*x^-1)^4, 14, 1, 0, g1^-1 ], [ x^-1, x^-1, 14, 1, 1, g1 ], [ (y^-1*x^-1)^5, (y^-1*x^-1)^5, 17, 2, 0, g2^-1 ], [ (y*x)^6*y, (y*x)^6*y, 17, 2, 1, g2 ], [ (y^-1*x^-1)^6, (y^-1*x^-1)^6, 20, 3, 0, g3^-1 ], [ (y*x)^5*y, (y*x)^5*y, 20, 3, 1, g3 ], [ (y^-1*x^-1)^7, (y^-1*x^-1)^7, 23, 4, 0, g4^-1 ], [ (y*x)^4*y, (y*x)^4*y, 23, 4, 1, g4 ] ]
The basic file structure of a HyperCell model graph file is as follows (with placeholders marked by <...>):
HyperCells HCM version 1.0 [ <r>, <q>, <p> ] [ <relators> , <center> ] [ g1, g2, ... ] -> <generators> <TDGAMMA> [ <TGGx>, <TGGy>, <TGGz> ] <model type> <vertices> <edges> <edge translations> <faces>
The first line specifies the file format and its version.
The second line gives the signature (r,q,p) of the underlying triangle group \Delta(r,q,p) and the third line specifies the triangle-group quotient \Delta^+(r,q,p)/\Gamma by the list of relators <relators> given as elements of the proper triangle group \Delta^+ with generators x, y, z and the center of the cell given as the integer <center> with 1, 2, 3 corresponding to the center lying at the vertex x, y, z of the fundamental Schwarz triangle, respectively.
The fourth line specifies the generators of the translation group \Gamma. To the left of -> are the generators of the finitely presented group and to the right the list <generators> of their images in the proper triangle group \Delta^+ in terms of the generators x, y, z.
The fifth line gives the transversal T_{\Delta^+}(\Gamma) as a list <TDGAMMA> of elements of the proper triangle group \Delta^+ and the sixth line specifies the transversals T_{G^+}(G_w^+) as lists <TGGw> with w=x,y,z (w=x,y,z) of preimages of elements of the quotient group G^+\cong\Delta^+/\Gamma under the quotient homomorphism \Delta^+\to G^+, i.e., in terms of the generators x, y, z.
The seventh line specifies the model type. The following model types are currently implemented:
Derived graph: general model graph with vertices, edges, and faces derived from the cell graph. The model type is given as [ "VEF", [ vfs, efs, ffs ] ], where vfs, efs, and ffs are lists of integers between 1 and 3 indicating the vertex types, edge types, and face types, respectively. See TGCellModelGraph (4.2-1) for details.
Tessellation graph: tessellation model graph derived from a cell graph of a triangle group. The model type is given as [ "TESS", [ p, q ], [ "VEF", [ [ 2 ], [ 1 ], [ 3 ] ] ] ] for the \{p,q\} tessellation and as [ "TESS", [ q, p ], [ "VEF", [ [ 3 ], [ 1 ], [ 2 ] ] ] ] for the (dual) \{q,p\} tessellation. See TessellationModelGraph (4.2-2) for details.
Kagome graph: kagome model graph derived from a cell graph of a triangle group. The model type is given as [ "KAGOME", p, [ "VEF", [ [ 1 ], [ 2 ], [ 2, 3 ] ] ] ]. See KagomeModelGraph (4.2-4) for details.
Lieb graph: Lieb model graph derived from a cell graph of a triangle group. The model type is given as [ "LIEB", [ p, q ], [ "VEF", [ [ 1, 2 ], [ ], [ 3 ] ] ] ] for the \{p,q\} tessellation and as [ "LIEB", [ q, p ], [ "VEF", [ [ 1, 3 ], [ ], [ 2 ] ] ] ] for the (dual) \{q,p\} tessellation. See LiebModelGraph (4.2-5) for details.
Lines eight to eleven specify the vertices, edges, edge translations and faces of the cell:
The vertices are given as a list <vertices> with entries of the form [ w, gi ], where w is 1, 2, 3 corresponding to the vertex x, y, z of the fundamental Schwarz triangle and gi is the position of the element in the transversal T_{G^+}(G_w^+).
The edges are given as a list <edges> with entries of the form [ v1, v2, tag ], where v1 and v2 are the positions of the vertices in <vertices> connected by the edge, and tag is a unique identifier of the edge. More details on the tags are given in the methods for constructing different models, see 4.2.
The edge translations, i.e., the translations relating the unit cell containing the first vertex (v1) to the unit cell containing the second (v2) are given as a list <edge translations> of elements of the finitely presented group, i.e., in terms of the generators given to the left of -> in the third line. If v1 corresponds to the vertex (w, [g_{u_w}]_w) and v2 to the vertex (w', [g_{\tilde{u}_{w'}}]_{w'}), then gam corresponds to the translation \gamma such that
g_{\tilde{u}_{w'}} = g_{u_w}\gamma.
The faces are given as a list <faces> with entries of the form [ edge, orientation ], where edge is the position of the edge in <edges> and orientation is 1 or -1 depending on whether the edge is traversed in the same or opposite direction as the edge given in <edges>.
The full file structure of a HyperCell cell graph file thus is as follows:
HyperCells HCM version 1.0 [ <r>, <q>, <p> ] [ [ <rel1>, <rel2>, ... ], <center> ] [ g1, g2, ... ] -> [ <generator1>, <generator2>, ... ] [ <d1>, <d2>, ... ] [ [ <gx1>, <gx2>, ... ], [ <gy1>, <gy2>, ... ], [ <gz1>, <gz2>, ... ] ] <model type> [ [ w, gi ], ... ] [ [ v1, v2, tag ], ... ] [ [ gam1, gam2, ... ], ... ] [ [ edge, orientation ], ... ]
A complete example for the {8,8} tessellation graph centered at the z vertex on the cell defined by the quotient T2.6 of \Delta(2,8,8) is given below:
HyperCells HCM version 1.0 [ 2, 8, 8 ] [ [ x^2, x*y*z, x*z*y, y^3*z^-1 ], 3 ] [ g1, g2, g3, g4 ] -> [ z^-1*x^-1*y*z^-1*y*z*y*x^-1, z^-1*x^-1*y*z^-1*y*z*y*x^-1*y*x*z, z*y*(x^-1*z^-1)^2*y^3*x*z, y^-1*z*y^-1*x*z ] [ 1, y^-1*x^-1, (y^-1*x^-1)^2, (y^-1*x^-1)^3, (y^-1*x^-1)^4, (y^-1*x^-1)^5, (y^-1*x^-1)^6, (y^-1*x^-1)^7 ] [ [ 1, (y^-1*x^-1)^3, y^-1*x^-1, (y^-1*x^-1)^2 ], [ 1 ], [ 1 ] ] [ "TESS", [ 8, 8 ], [ "VEF", [ [ 3 ], [ 1 ], [ 2 ] ] ] ] [ [ 3, 1 ] ] [ [ 1, 1, [ 1, [ [ 1, 1 ], 1, 5 ] ] ], [ 1, 1, [ 1, [ [ 1, 2 ], 4, 8 ] ] ], [ 1, 1, [ 1, [ [ 1, 3 ], 2, 6 ] ] ], [ 1, 1, [ 1, [ [ 1, 4 ], 3, 7 ] ] ] ] [ g1, g4, g2, g3 ] [ [ [ 1, -1 ], [ 2, -1 ], [ 4, 1 ], [ 3, -1 ], [ 1, 1 ], [ 2, 1 ], [ 4, -1 ], [ 3, 1 ] ] ]
The basic file structure of a HyperCell supercell model graph file is as follows (with placeholders marked by <...>):
HyperCells HCS version 1.0 [ <r>, <q>, <p> ] [ <primitive cell relators> , <supercell relators>, <center> ] [ g1, g2, ... ] -> <pc generators> <TDGAMMApc> [ <TGpcGx>, <TGpcGy>, <TGpcGz> ] [ g1, g2, ... ] -> <sc generators> [ g1, g2, ... ] -> <sc generators in terms of pc generators> <TGAMMApcGAMMAsc> <TDGAMMAsc> [ <TGscGx>, <TGscGy>, <TGscGz> ] <model type> <vertices> <edges> <edge translations> <faces>
The first line specifies the file format and its version.
The second line gives the signature (r,q,p) of the underlying triangle group \Delta(r,q,p) and the third line specifies the triangle-group quotient \Delta^+(r,q,p)/\Gamma_\mathrm{pc} and \Delta^+(r,q,p)/\Gamma_\mathrm{sc} by the list of relators <primitive cell relators> and <supercell relators>, respectively, given as elements of the proper triangle group \Delta^+ with generators x, y, z and the center of the cell given as the integer <center> with 1, 2, 3 corresponding to the center lying at the vertex x, y, z of the fundamental Schwarz triangle, respectively.
The fourth line specifies the generators of the translation group \Gamma_\mathrm{pc} of the primitive cell. To the left of -> are the generators of the finitely presented group and to the right the list <generators> of their images in the proper triangle group \Delta^+ in terms of the generators x, y, z.
The fifth line gives the transversal T_{\Delta^+}(\Gamma_\mathrm{pc}) as a list <TDGAMMApc> of elements of the proper triangle group \Delta^+ and the sixth line specifies the transversals T_{G^+_\mathrm{pc}}(G_w^+) as lists <TGpcGw> with w=x,y,z (w=x,y,z) of preimages of elements of the quotient group G^+_\mathrm{pc}\cong\Delta^+/\Gamma_\mathrm{pc} under the quotient homomorphism \Delta^+\to G^+_\mathrm{pc}, i.e., in terms of the generators x, y, z.
Lines seven, ten, and eleven give the same information as lines three, four, and five, respectively, but for the supercell (G^+_\mathrm{sc}, \Gamma_\mathrm{sc}) instead of the primitive cell. The eighth line specifies the homomorphism embedding the normal subgroup \Gamma_\mathrm{sc} in \Gamma_\mathrm{pc} with the generators of the finitely presented group \Gamma_\mathrm{sc} to the left of -> and their images in the finitely presented group \Gamma_\mathrm{pc} to the right. The ninth line gives the corresponding transversal T_{\Gamma_\mathrm{pc}} (\Gamma_\mathrm{sc}) as a list <TGAMMApcGAMMAsc> of elements of the triangle group \Delta^+.
The twelfth line specifies the model type. The following model types are currently implemented:
Derived graph: general model graph with vertices, edges, and faces derived from the cell graph. The model type is given as [ "VEF", [ vfs, efs, ffs ] ], where vfs, efs, and ffs are lists of integers between 1 and 3 indicating the vertex types, edge types, and face types, respectively. See TGCellModelGraph (4.2-1) for details.
Tessellation graph: tessellation model graph derived from a cell graph of a triangle group. The model type is given as [ "TESS", [ p, q ], [ "VEF", [ [ 2 ], [ 1 ], [ 3 ] ] ] ] for the \{p,q\} tessellation and as [ "TESS", [ q, p ], [ "VEF", [ [ 3 ], [ 1 ], [ 2 ] ] ] ] for the (dual) \{q,p\} tessellation. See TessellationModelGraph (4.2-2) for details.
Kagome graph: kagome model graph derived from a cell graph of a triangle group. The model type is given as [ "KAGOME", p, [ "VEF", [ [ 1 ], [ 2 ], [ 2, 3 ] ] ] ]. See KagomeModelGraph (4.2-4) for details.
Lieb graph: Lieb model graph derived from a cell graph of a triangle group. The model type is given as [ "LIEB", [ p, q ], [ "VEF", [ [ 1, 2 ], [ ], [ 3 ] ] ] ] for the \{p,q\} tessellation and as [ "LIEB", [ p, q ], [ "VEF", [ [ 1, 3 ], [ ], [ 2 ] ] ] ] for the (dual) \{q,p\} tessellation. See LiebModelGraph (4.2-5) for details.
Lines thirteen to sixteen specify the vertices, edges, edge translations and faces of the cell:
The vertices are given as a list <vertices> with entries of the form [ w, gi ], where w is 1, 2, 3 corresponding to the vertex x, y, z of the fundamental Schwarz triangle and gi is the position of the element in the transversal T_{G^+}(G_w^+).
The edges are given as a list <edges> with entries of the form [ v1, v2, tag ], where v1 and v2 are the positions of the vertices in <vertices> connected by the edge, and tag is a unique identifier of the edge. The tag tag is of the form [ v1pc, v2pc, tagpc ] with v1pc, v2pc the positions of the vertices in the primitive cell, and tagpc the tag of the edge in the primitive cell (see 4.2). Thus, tag specifies (up to the translation) the edge in the primitive cell.
The edge translations, i.e., the translations relating the unit cell containing the first vertex (v1) to the unit cell containing the second (v2) are given as a list <edge translations> of elements of the finitely presented group, i.e., in terms of the generators given to the left of -> in the third line. If v1 corresponds to the vertex (w, [g_{u_w}]_w) and v2 to the vertex (w', [g_{\tilde{u}_{w'}}]_{w'}), then gam corresponds to the translation \gamma such that
g_{\tilde{u}_{w'}} = g_{u_w}\gamma.
The faces are given as a list <faces> with entries of the form [ edge, orientation ], where edge is the position of the edge in <edges> and orientation is 1 or -1 depending on whether the edge is traversed in the same or opposite direction as the edge given in <edges>.
The full file structure of a HyperCell cell graph file thus is as follows:
HyperCells HCS version 1.0 [ <r>, <q>, <p> ] [ [ <rel1>, <rel2>, ... ], [ <rel1>, <rel2>, ... ], <center> ] [ g1, g2, ... ] -> [ <pc generator1>, <pc generator2>, ... ] [ <d1>, <d2>, ... ] [ [ <gx1>, <gx2>, ... ], [ <gy1>, <gy2>, ... ], [ <gz1>, <gz2>, ... ] ] [ g1, g2, ... ] -> [ <sc generator1>, <sc generator2>, ... ] [ eta1, eta2, ... ] [ <dsc1>, <dsc2>, ... ] [ [ <gscx1>, <gscx2>, ... ], [ <gscy1>, <gscy2>, ... ], ... ] <model type> [ [ w, gi ], ... ] [ [ v1, v2, s ], ... ] [ [ gam1, gam2, ... ], ... ] [ [ edge, orientation ], ... ]
A complete example for the {8,8} tessellation graph centered at the z vertex on the cell defined by the quotient T2.6 of \Delta(2,8,8) extended to the supercell defined by the quotient T3.11 is given below:
HyperCells HCS version 1.0 [ 2, 8, 8 ] [ [ x^2, x*y*z, x*z*y, y^3*z^-1 ], [ x^2, x*y*z, x*z*y, y^-8 ], 3 ] [ g1, g2, g3, g4 ] -> [ z^-1*x^-1*y*z^-1*y*z*y*x^-1, z^-1*x^-1*y*z^-1*y*z*y*x^-1*y*x*z, z*y*(x^-1*z^-1)^2*y^3*x*z, y^-1*z*y^-1*x*z ] [ 1, y^-1*x^-1, (y^-1*x^-1)^2, (y^-1*x^-1)^3, (y^-1*x^-1)^4, (y^-1*x^-1)^5, (y^-1*x^-1)^6, (y^-1*x^-1)^7 ] [ [ 1, (y^-1*x^-1)^3, y^-1*x^-1, (y^-1*x^-1)^2 ], [ 1 ], [ 1 ] ] [ g1, g2, g3, g4, g5, g6 ] -> [ x*y^-1*z^-1, y*x*z, z*y^-1*z^-1*y, y^2*x*z*y^-1, y^-1*z*y^-1*z^-1*y^2, y^3*x*z*y^-2 ] [ g1, g2, g3, g4, g5, g6 ] -> [ g1^-1*g4^-1, g1^-1*g2, g3*g4^-1, g1^-1*g2*g4^-1*g1^-1*g2*g3^-1*g4*g1*g2^-1*g1, g4*g1^-1*g4^-1*g3*g2^-1*g1, g1^-1*g2*g3^-1*g4*g2^-1*g1 ] [ 1, y^3*z^-1 ] [ 1, x, y, y^-1, z, z^-1, y^2, y*z^-1, y^-2, y^-1*z, y^3, y^2*z^-1, y^-3, y^-2*z, y^4, y^3*z^-1 ] [ [ 1, y, y^-1, y^2, y^-2, y^3, y^-3, y^4 ], [ 1, x ], [ 1, x ] ] [ "TESS", [ 8, 8 ], [ "VEF", [ [ 3 ], [ 1 ], [ 2 ] ] ] ] [ [ 3, 1, 1 ], [ 3, 1, 2 ] ] [ 1, x ] [ [ 1, 2, [ 1, 1, [ 1, [ [ 1, 1 ], 1, 5 ] ] ] ], [ 1, 2, [ 1, 1, [ 1, [ [ 1, 2 ], 4, 8 ] ] ] ], [ 1, 2, [ 1, 1, [ 1, [ [ 1, 3 ], 2, 6 ] ] ] ], [ 1, 2, [ 1, 1, [ 1, [ [ 1, 4 ], 3, 7 ] ] ] ], [ 2, 1, [ 1, 1, [ 1, [ [ 1, 1 ], 1, 5 ] ] ] ], [ 2, 1, [ 1, 1, [ 1, [ [ 1, 2 ], 4, 8 ] ] ] ], [ 2, 1, [ 1, 1, [ 1, [ [ 1, 3 ], 2, 6 ] ] ] ], [ 2, 1, [ 1, 1, [ 1, [ [ 1, 4 ], 3, 7 ] ] ] ] ] [ 1, g4^-1*g6*g2, g2, g4^-1*g2, g3*g5^-1*g3^-1*g1^-1*g2^-1*g4^-1*g6^-1*g4, g1^-1, g3*g5^-1*g1^-1, g3*g1^-1 ] [ ]
In addition, the following file formats are introduced by the HyperCells package:
HyperCell quotient sequences structure (*.hcqs) for storing the adjacency matrices, i.e., TGQuotientSequencesStructure objects (see 6.2)
HyperCell point-group matrices (*.hcpgm) for storing the point-group matrices, i.e., PGMatrices objects (see 7.3)
These file formats are described in detail below.
The basic file structure of a HyperCell quotient sequences structure file is as follows (with placeholders marked by <...>):
HyperCells HCQS version 1.0 [ <r>, <q>, <p> ] <boundByGenus> <quotients> <mirror symmetries> <sparse> <adjacency matrix>
The first line specifies the file format and its version. The second line gives the signature (r,q,p) of the underlying triangle group \Delta(r,q,p). The third line specifies the upper bound for triangle group quotients with genus smaller than <boundByGenus>, and the forth line, <quotients>, the corresponding list of triangle group quotients considered.
Line five, <mirror symmetries>, specifies an ordered binary list of ones and zeros, 1 if quotients in <quotients> have mirror symmetries 0 otherwise.
The sixth line <sparse> specifies if line seven, the adjacency matrix <adjacency matrix>, is sparsely represented true, or dense false.
The full file structure of a HyperCell quotient sequences structure file thus is as follows:
HyperCells HCQS version 1.0 [ <r>, <q>, <p> ] <boundByGenus> [ [ <genus1>, <number1> ], [ <genus2>, <number2> ], ... ] [ b1, b2, ... ] <sparse> [ entry1, entry2, ... ]
A complete example for a quotient sequences structure with a dense adjacency matrix for the \Delta(2,8,8) triangle group and triangle group quotients with genus smaller than 12 is given below:
HyperCells HCQS version 1.0 [ 2, 8, 8 ] 12 [ [ 2, 6 ], [ 3, 10 ], [ 3, 11 ], [ 5, 12 ], [ 5, 13 ], [ 9, 19 ], [ 9, 20 ], [ 9, 21 ], [ 9, 22 ], [ 9, 23 ], [ 10, 22 ], [ 11, 12 ], [ 11, 13 ] ] [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0 ] false [ [ 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ], [ 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ])
A complete example for a sparse representation of the adjacency matrix of the quotient sequences structure for the \Delta(2,8,8) triangle group and triangle group quotients with genus smaller than 12 is given below:
HyperCells HCQS version 1.0 [ 2, 8, 8 ] 12 [ [ 2, 6 ], [ 3, 10 ], [ 3, 11 ], [ 5, 12 ], [ 5, 13 ], [ 9, 19 ], [ 9, 20 ], [ 9, 21 ], [ 9, 22 ], [ 9, 23 ], [ 10, 22 ], [ 11, 12 ], [ 11, 13 ] ] [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0 ] true [ [ [ 1, 3 ], 1 ], [ [ 1, 5 ], 1 ], [ [ 1, 7 ], 1 ], [ [ 1, 9 ], 1 ], [ [ 1, 11 ], 1 ], [ [ 1, 13 ], 1 ], [ [ 2, 5 ], 1 ], [ [ 2, 7 ], 1 ], [ [ 2, 9 ], 1 ], [ [ 2, 12 ], 1 ], [ [ 3, 5 ], 1 ], [ [ 3, 7 ], 1 ], [ [ 3, 9 ], 1 ], [ [ 3, 13 ], 1 ], [ [ 4, 6 ], 1 ], [ [ 4, 8 ], 1 ], [ [ 4, 9 ], 1 ], [ [ 5, 7 ], 1 ], [ [ 5, 9 ], 1 ] ])
The basic file structure of a HyperCell point-group matrices file is as follows (with placeholders marked by <...>):
HyperCells HCPGM version 1.0 [ <r>, <q>, <p> ] <quotient> <sparse> <abcPGMats> <symPGMats>
The first line specifies the file format and its version. The second line gives the signature (r,q,p) of the underlying triangle group \Delta(r,q,p) and the third line gives the triangle group quotient. The forth line <sparse> specifies if line five and six containing the point-group matrices <abcPGMats> and <symPGMats>, respectively, are sparsely represented true, or dense false.
The full file structure of a HyperCell point-group matrices file thus is as follows:
HyperCells HCPGM version 1.0
[ <r>, <q>, <p> ]
[ <quotientName>, <order>, <genus>, <actionType>, <rels> ]
<sparse>
[ [ "a", <pgMatA> ], [ "b", <pgMatB> ], [ "c", <pgMatC> ] ]
[ [ [ <symName1>, <sym1> ], <pgMat1> ], [ [ <symName2>, <sym2> ], <pgMat2> ],
... ]
A complete example for the point-group matrices for the \Delta(2,8,8) triangle group and triangle group quotient T2.6 with symmetry elements a, b, c as generators of \Delta(2,8,8) and x, y, z as generators of \Delta^+(2,8,8) which we choose to denote as "X", "Y", "Z":
HyperCells HCPGM version 1.0
[ 2, 8, 8 ]
[ [ 2, 6 ], 8, 2, "Action reflexible [m,n]",
"[ x^2, x * y * z, x * z * y, y^3 * z^-1 ]" ]
false
[ [ "a",
[ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ -1, -1, -1, 0 ], [ 0, 0, 0, 1 ] ] ],
[ "b",
[ [ 0, -1, 0, 0 ], [ -1, 0, 0, 0 ], [ 1, 1, 1, 0 ], [ 0, 0, 0, -1 ] ] ],
[ "c",
[ [ -1, 0, 0, 0 ], [ 1, 1, 1, 1 ], [ 0, 0, 0, -1 ], [ 0, 0, -1, 0 ] ] ] ]
[ [ [ "X", "x" ],
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ] ],
[ [ "Y", "y" ],
[ [ -1, -1, -1, -1 ], [ 1, 0, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 1, 0 ] ] ],
[ [ "Z", "z" ],
[ [ 0, -1, 0, 0 ], [ 0, 0, -1, 1 ], [ 0, 0, 0, -1 ], [ 1, 1, 1, 0 ] ] ] ]
A complete example for a sparse representation of the point-group matrices for the \Delta(2,8,8) triangle group and triangle group quotient T2.6 with symmetry elements a, b, c as generators of \Delta(2,8,8) and x, y, z as generators of \Delta^+(2,8,8) which we choose to denote as "X", "Y", "Z":
HyperCells HCPGM version 1.0
[ 2, 8, 8 ]
[ [ 2, 6 ], 8, 2, "Action reflexible [m,n]",
"[ x^2, x * y * z, x * z * y, y^3 * z^-1 ]" ]
true
[ [ "a", [ [ [ 1, 2 ], 1 ], [ [ 2, 1 ], 1 ], [ [ 3, 1 ], -1 ],
[ [ 3, 2 ], -1 ], [ [ 3, 3 ], -1 ], [ [ 4, 4 ], 1 ] ] ],
[ "b", [ [ [ 1, 2 ], -1 ], [ [ 2, 1 ], -1 ], [ [ 3, 1 ], 1 ],
[ [ 3, 2 ], 1 ], [ [ 3, 3 ], 1 ], [ [ 4, 4 ], -1 ] ] ],
[ "c", [ [ [ 1, 1 ], -1 ], [ [ 2, 1 ], 1 ], [ [ 2, 2 ], 1 ],
[ [ 2, 3 ], 1 ], [ [ 2, 4 ], 1 ], [ [ 3, 4 ], -1 ],
[ [4, 3 ], -1 ] ] ] ]
[ [ [ "X", "x" ], [ [ [ 1, 1 ], -1 ], [ [ 2, 2 ], -1 ], [ [ 3, 1 ], -1 ],
[ [ 3, 2 ], -1 ], [ [ 3, 3 ], -1 ], [ [ 4, 4 ], -1 ] ] ],
[ [ "Y", "y" ], [ [ [ 1, 1 ], -1 ], [ [ 1, 2 ], -1 ], [ [ 1, 3 ], -1 ],
[ [ 1, 4 ], -1 ], [ [ 2, 1 ], 1 ], [ [ 3, 1 ], 1 ],
[ [ 3, 2 ], 1 ], [ [ 3, 3 ], 1 ], [ [ 3, 4 ], -1 ],
[ [ 4, 3 ], 1 ] ] ],
[ [ "Z", "z" ], [ [ [ 1, 2 ], -1 ], [ [ 2, 1 ], -1 ], [ [ 2, 2 ], -1 ],
[ [ 2, 3 ], -1 ], [ [ 2, 4 ], 1 ], [ [ 3, 4 ], -1 ],
[ [ 4, 1 ], 1 ], [ [ 4, 2 ], 1 ], [ [ 4, 3 ], 1 ] ] ] ]
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