Coset expansion

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,... 

D2 has the invariant subgroup D of index 2, with the coset expansion... 

Example 16.4-2 Write a coset expansion of the point group P, for the strictly 2-D square lattice, on P(k), where k is the vector from the origin to point Z in Figure 16.14. 

The final step going from the small IRs of the little group G(k) to the IRs of G requires the theory of induced representations (Section 4.8). At a particular k in the representation domain, the left coset expansion of G on the little group G(k) is... 

Let w be the non-lattice translation vector that belongs to Ru in the coset expansion, eq. (2). Then... 

For the space group 225 (Fmim or Ojj) write down the coset expansion of the little group G(k) on T. Hence write down an expression for the small representations. State the point group of the k vector P(k) at the symmetry points L(/4 V-i Vi) and X(q a 2a). Work out also the Cartesian coordinates of L and X. Finally, list the space-group representations at L and X. 

The opposite process to subduction is induction. Here, we start from an irrep in a subgroup H. By coset expansion, this subgroup is put on an orbit inside a higher symmetry group. This leads to an extension of the function space and generates... 

Consider a subgroup H cG such that G / H =2. Then the coset expansion of G will be limited to only two cosets ... 

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