Chain Subduction Criterion

The Chain Subduction Criterion is final condition for select low-symmetry phase groups for a given irrep, or alternatively, for the symmetry breaking representation for a given low-symmetry phase group. The condition can be formulated as follows:

Let H₁ be a subgroup of G₀ and let H₂ be a maximal subgroup of H₁:

$$H_2 \subset H_1 \subseteq G_0$$ (26)

Consider the subduction of irrep D_G0^j to both subgroups H₁ and H₂:

$$D_{G_0}^j = \sum_m \left( D_{D_0}^j \\ gt \bigl\vert \\ gt D... ...( D_{D_0}^j \\ gt \bigl\vert \\ gt D_{H_2}^s \right) D_{H_2}^s,$$ (27)

where for the subduction coefficients of the identity, the following relations holds:  

$$\left( D_{G_0}^j \\ gt \bigl\vert \\ gt D_{H_2}^1 \right) \geq \left( D_{G_0}^j \\ gt \bigl\vert \\ gt D_{H_1}^1 \right).$$ (28)

If the subduction multiplicity coefficients in (28) are equal for H₁ and for H₂, then the subgroup H₂ is eliminated as a possible isotropy group.

The colour group reformulation: If the multiplicity coefficients of D_G0^j are equal in the permutational representations D_G0^H₁ and D_G0^H₂ of both groups, the colour group with greater number of colours, do not corresponds to PT.