A note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensions

It is shown that there are 51 types of physical quantities in arbitrary dimensions with distinct transformations by wedge reversion symmetry. In the paper by Gopalan [(2020). Acta Cryst. A76, 318–327] only 41 types were enumerated.

Let I 1 and I 2 be two different antisymmetries (any out of the seven). If the action of both I 1 and I 2 on some multivector is mixed then the action of I 1 Á I 2 on this multivector can be even, odd or mixed. Specifying the action of only three antisymmetries (especially 1, 1 0 and 1 y ) on a multivector, as was considered by Gopalan (2020), is not sufficient to obtain the result of the action of the remaining four antisymmetries; see a simplified example with three antisymmetries in Table 1. This has led to a clustering of different multivector types into one type in Table 1 of Gopalan (2020): the types numbered 16,19,22,25,28 and 31 should be separated into two types each and type 38 into five types. This gives in total ten new multivector types which were not given by Gopalan (2020), as shown in Table 2 for all seven antisymmetries. New labels for the X, Y, Z multivectors are proposed in Table 2 in a coherent notation, which uses four out of the eight principal multivectors. An extended version of Table 2 with grades and examples of multivectors is given in Table 3, which is the final table for these new results, with all 51 multivector types describing the action of all seven antisymmetries, given in the same layout as Table 1 of Gopalan (2020). Table 1 An example of the action of 1, 1 0 and 1 0 antisymmetries on several multivectors.
The action of antisymmetries 1 and 1 0 on S+V 0 and S+V 0 +V gives mixed results, while the action of the product antisymmetry 1 0 can be odd or mixed.
Odd Even Mixed Mixed Table 2 Splitting of multivector types, with the left-hand side displaying the number, stabilizer subgroup, label and action of 1, 1 0 and 1 y antisymmetries as given by Gopalan (2020), and the right-hand side displaying the number, stabilizer subgroup, label and action of all antisymmetries as presented in this work.
Considering the action of all antisymetries leads to splitting of multivector types. The last three rows describe the new labels of the X, Y and Z multivector types, without splitting. An extended version of this table with grades and examples of multivectors is available in the supporting information.
Work of Gopalan (2020) This paper Action of Action of Table 3 Classification of extended multivector types for physical properties using the same notation as in Table 1 of Gopalan (2020).
The actions of all seven generalized inversions and grades are given explicitly. Entries in bold in columns 1 and 2 are the eight principal multivector types. Note that the last column contains sums (not products) of principal multivectors, but the '+' signs are omitted.