Coset representatives

The corresponding left coset expansion with the same coset representative is... 

Note that the elements of G do not have to appear in exactly the same order in the left and right coset expansions. This will only be so if the coset representatives commute with every element of H. All that is necessary is that the two lists of elements evaluated from the coset expansions both contain each element of G once only. It should be clear from eqs. (5) and (6) that H gr = gr H, where H= P0 Pi P2 and gr is P4. An alternative way of testing for invariance is to evaluate the transforms of H. For example,... 

Consequently, Hi is not an invariant subgroup. For H to be an invariant subgroup of G, right and left cosets must be equal for each coset representative in the expansion of G. 

Consider the group G = H, AH that contains unitary H and antiunitary AH operators. H is necessarily an invariant subgroup of G of index 2 and AH is a coset of H with coset representative A (which may be any one of the antiunitary operators of G) so that... 

Each term in square brackets in eq. (35) is itself a set of elements, being T multiplied by the coset representative (f w). Therefore F is isomorphous with the point group P. The kernel... 

Exercise 16.6-2 Does the set of coset representatives in (15) form a group ... 

Wi = Rn and W2 = Ri2, where Rn, Ri2 are representations of the abstract group G32 which is isomorphous with + (k). Column headings for the classes are the coset representatives ( A w). Time-reversal symmetry is of type a for both representations. 

Of these possibilities (as Table 17.17 shows) only the pair da = E,db=E are double coset representatives that satisfy (eq. 9),... 

In this appendix we prove the corrolary to the theorem of Hall that the orbit of the cosets of a subgroup H of group G can only be doubly transitive for H a maximal subgroup of G. To this end we consider a further subgroup S c H, and examine if the orbit of cosets of S can be doubly transitive. Let gr and hp denote cosets representatives of H in G, and S in H resp., i.e. ... 

As we have seen in the previous chapter, the coset representatives each address a copy of site a), which we shall label as (ic). The site group that stabilizes this site is isomorphic to and is denoted by thus have the following mappings ... 

Let operator g = t R correspond to the coset representative t R in the coset decomposition of the crystal structure space group G over translation subgroup T (see (2.15)). In this notation the density matrix point symmetry can be written in the form... 

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