PR and TR representations for order 5 LS’s (par. 3.4)
We wish to determine the number of LS’s representing the full order 5 set if we count all point symmetry related squares as one (PR), as well as the similar number when counting all translation-related members as one (TR).
We will collect the dihedral-group related LS’s and the translation related LS’s in distinct “manyfolds”, symbolized in Chapter 2 by ( ) and [ ] respectively, but now for each structure type. The numbers we are looking for are the numbers of both types of these manyfolds summed over all 18 structure types.
To show how to do this we collect the (generally) 200 LS’s for a structure type on multiplication tables IIa and IIb). with dihedral group operations index l running left to right from 0 to 7 and 25 translations with indices hk running from (00) to (44) downwards. Assessing (in)equalities of the products is facilitated by expressing them as HP 48 checksums (par. 3.3 and 4.4).
As an example we will obtain the PR-contribution for structure type, 6.1, starting out with the preferred member for this type given in Table 3.2.1. We first prepare column hk0 by letting all hk translations act on structure type 6.1 and next obtain rows hkn horizontally by applying the dihedral group operations on each member of the hk0 series. The PR-number contribution of each independent row (manyfold) is 1 and the full contribution of the entire spreadsheets of the symmetry-free structures 4 and 11 would have been 25 (the default multiplicity). In our spread sheet for type 6.1 we observe that the content of some of the rows is equal except for order. The numbers of different rows (the actual multiplicity) goes down to 1 in structure types 1 and 14. The number in our case , 6.1, is 15 (see subscript of Table 3.4.1a), given already in List 3.2.2) .
To obtain the TR-contribution we proceeded in the reverse way, preparing rows l00 first and obtaining the translation related LS’s in columns lhk.
Table IIIa. Sections of the multiplication table with intermediate results hk0 ( column 0) when translations hk have acted on a member of structure type 6.1 shown in Table 3.2.1 and with final results hkn when next point group operations act on the intermediate results. Data are HP48 checksums of the resulting LS’s.
We see that the content of each row pair (hk-n, kh-n, three pairs shown) is identical except for the order; the pair can be represented by one LS. There are 10 pairs of such rows; this brings, together with the contribution of five hh-n rows (two rows shown), the contribution of this structure on the PG- number to 15 LS. Corresponding contributions of the other structure types are given in List 3.2.2 as well as their sum: 192.
Let us next prepare 00-l first and compute the checksums of the hk-l products:
Table IIIb. Sections of the multiplication table with intermediate results l00(first row) when point group movements n act on the member of structure type 6.1 shown in table 3.2.1 and with final results after action of the translation operations on the intermediate results, obtaining the same abbreviated HP 48 checksums for the LS’s as in Table IIIa but in different order.
Checksums along the 00-n series are clearly two by two identical for structure type 6.1, (0,6), (1,4), etc., and the same will hold of course for the corresponding columns, meaning we have only 4 distinct columns so the contribution of this structure type to the TG-number is 4 . See List 3.2.2 for the contribution of the other structure types and their sum: 60.
Note that successive point group – and translation operations are non Abelian: the products hk-n and n-hk are generally unequal, as shown in the Tables with 20-2 and 2-20.