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Symmetry Analysis

The fundamental properties of liquid crystals are governed by their symmetries, which dictate the possible order parameters and their associated transformations.
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The statement of the theorem

Let GG be the point group symmetry of the liquid crystal phase. The physical properties are invariant under the action of GG. The order parameters ηi\eta_{i} must transform according to the irreducible representations (irreps) Γk\Gamma_{k} of GG. The free energy density ff must be a scalar invariant under the group action: \nf(ηi)=f(g⋅ηi)∀g∈Gf(\eta_{i}) = f(g \cdot \eta_{i}) \quad \forall g \in G \nThis requires that ff can be expressed as a linear combination of the basis invariants of the group.