Sequences of triangle-group quotients G^{(n)} are used to approximate the thermodynamic limit, either in real space [LP23], or in reciprocal space [LMB23]. They are constructed as quotients of the (proper) triangle group \Delta^+ by translation groups \Gamma^{(n)}\triangleleft\Delta^+ such that
\Delta^+ = \Gamma^{(0)} \triangleright \Gamma^{(1)} \triangleright \Gamma^{(2)} \triangleright \cdots
and \bigcap_{n\geq 0} \Gamma^{(n)} = {1}.
To check whether a given sequence of triangle-group quotients is valid, i.e., whether the corresponding translation groups form a normal sequence as described above, the following functio can be used:
‣ IsTGQuotientSequence( sequence ) | ( function ) |
returns true if the given list of TGQuotient objects (see 2.3) is a valid sequence of quotient groups of the same triangle group, i.e., if the corresponding translation groups form a normal sequence as described above.
While such sequences can be found in the library of triangle-group quotients described in 2.4, an exhaustive search based on low-index normal subgroups such as the one that that library is based on is not efficient for producing long sequences directly. However, given a normal sequence of translation groups and one additional translation group not part of the sequence, it is possible to extend the sequence by forming intersections [Len23]. Let \Gamma^{(n)} be the last element of the sequence and \Gamma'\triangleleft\Delta^+ a translation group not part of the sequence, then
\Gamma^{(n+1)} = \Gamma^{(n)} \cap \Gamma'
is a normal subgroup of both \Gamma^{(n)} and \Delta^+ (but not necessarily a strict subgroup of \Gamma^{(n)}). This can be used to extend sequences of triangle-group quotients.
The HyperCells package implements the following functions to extend sequences:
‣ ExtendTGQuotientSequence( quotients, sequence ) | ( function ) |
Returns: sequence of triangle-group quotients as a list of TGQuotient objects.
Extends the sequence sequence using the list of additional quotients quotients, where both arguments are lists of TGQuotient objects (see 2.3). Intersections of the last element of the (extended) sequences are successively formed with the quotients in quotients.
‣ NextTGQuotientOptions( quotients, Q0 ) | ( function ) |
Returns: list of viable options of quotients in the form [ [ Q, ind ], ... ], where Q is a TGQuotient object and ind is the index of the intersection of the translation groups of Q and Q0 in the translation group of Q0.
Searches for viable options to extend a sequence ending with the quotient Q0 using the list of additional quotients quotients, where all quotients are given as TGQuotient objects (see 2.3).
‣ ExportTGQuotientList( list, path ) | ( function ) |
Exports the given list of TGQuotient objects (see 2.3) as a text file.
‣ ExportTGQuotientSequences( seqs, path ) | ( function ) |
Exports the given list of sequences, each a list of TGQuotient objects (see 2.3) as a text file.
‣ ImportTGQuotientList( input-stream ) | ( function ) |
‣ ImportTGQuotientListFromFile( path ) | ( function ) |
Returns: list of TGQuotient objects.
Import a list of TGQuotient objects (see 2.3) from the input-stream input-stream or from a file at path path.
‣ ImportTGQuotientSequences( input-stream ) | ( function ) |
‣ ImportTGQuotientSequencesFromFile( path ) | ( function ) |
Returns: list of list of TGQuotient objects.
Import a list of lists of TGQuotient objects (see 2.3) from the input-stream input-stream or from a file at path path.
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