Kernel-Core Criterion

The Kernel-Core Criterion (KCC) is an additional group-theoretical selection rule for the compatibility of irreps and subgroups. It is formulated as follows: If the PT between phases of symmetry G₀ and H₁ is associated with single irrep D_G0^j, then the kernel of this representation is equal to the core of subgroup H₁:

$$\mbox{ \rm Ker }D_{G_0}^j = \mbox{ \rm Core }H_1 = \bigcap_{g_i \in G_0} g_i H_1 g_i^{-1} = H.$$ (25)

The main power of this condition is that it fixes the minimal isotropy subgroup and this way the KCC ``cuts the tail'' and eliminates the chain of subgroups, which satisfy the Subduction Criterion.