(a) Underlying group action
Hermitian symmetry is not a geometric symmetry, but it is defined in terms of the action in reciprocal space of point group , i.e. , where e acts as I (the identity matrix) and acts as .
This group action on with will now be characterized by the calculation of the cocycle (Section 1.3.4.3.4.1) under the assumption that and are both diagonal. For this purpose it is convenient to associate to any integer vectorin the vector whose jth component is
Let , and hence . ThenhenceTherefore −e acts by
Hermitian symmetry is traditionally dealt with by factoring by 2, i.e. by assuming . If , then each is invariant under G, so that each partial vector (Section 1.3.4.3.4.1) inherits the symmetry internally, with a `modulation' by . The `multiplexing–demultiplexing' technique provides an efficient treatment of this singular case.
(b) Calculation of structure factors
The computation may be summarized as follows:where is the initial decimation given by , TW is the transposition and twiddlefactor stage, and is the final unscrambling by coset reversal given by .
(c) Calculation of electron densities
The computation may be summarized as follows:where is the decimation with coset reversal given by , TW is the transposition and twiddlefactor stage, and is the recovery in natural order given by .
(i) Decimation in time
The last transformation has a realvalued matrix, and the final result is realvalued. It follows that the vectors of intermediate results after the twiddlefactor stage are realvalued, hence lend themselves to multiplexing along the real and imaginary components of half as many general complex vectors.
Let the initial vectors be multiplexed into vectors[one for each pair ], each of which yields by F(M) a vectorThe realvaluedness of the may be used to recover the separate result vectors for and . For this purpose, introduce the abbreviated notationThen we may writeor, equivalently, for each ,Therefore and may be retrieved from Z by the `demultiplexing' formula:which is valid at all points where , i.e. whereDemultiplexing fails whenIf the pairs are chosen so that their members differ only in one coordinate (the jth, say), then the exceptional points are at and the missing transform values are easily obtained e.g. by accumulation while forming .
The final stage of the calculation is then
(ii) Decimation in frequency
The last transformation F(M) gives the realvalued results , therefore the vectors after the twiddlefactor stage each have Hermitian symmetry.
A first consequence is that the intermediate vectors need only be computed for the unique half of the values of , the other half being related by the Hermitian symmetry of .
A second consequence is that the vectors may be condensed into general complex vectors[one for each pair ] to which a general complex F(M) may be applied to yieldwith and realvalued. The final results can therefore be retrieved by the particularly simple demultiplexing formulae:

