Fully constrained mass matrix: Can symmetries alone determine the flavon vacuum alignments?

In the framework of the representation theory of finite groups, it was recently shown that a fully constrained complex-symmetric mass matrix can be conveniently mapped into a sextet of Sigma(72 x 3). In this paper, we introduce an additional flavor group X-24 in the model so that the vacuum alignment of the Sigma(72 x 3) sextet is determined not only by the symmetries of Sigma(72 x 3) but also by that of X-24. We define several flavons which transform as multiplets under Sigma(72 x 3) as well as X-24. The vacuum alignment of each of these flavons is obtained as a simultaneous invariant eigenstate of specific elements of the groups Sigma(72 x 3) and X-24; i.e., the vacuum alignment is fully determined by its residual symmetries. These flavons couple together uniquely resulting in the fully constrained sextet of Sigma(72 x 3). Through this work we propose a general formalism in which the flavor symmetry group (G(f)) is obtained as the direct product, G(f) = G(r) x G(x). Fermions transform nontrivially only under G(r) while they remain invariant under G(x). Flavons, on the other hand, transform nontrivially under both G(r) and G(x). The vacuum alignment of each flavon multiplet transforming irreducibly under G(r) x G(x) is uniquely identified by its corresponding residual symmetry (a subgroup of G(r) x G(x)). Several such flavons couple together to form an effective multiple of G(r) which remains invariant under G(x). This effective multiplet couples to the fermions.