Plaquette Invariants and the Flavour Symmetric Description of Quark and Neutrino Mixings

We present a complete set of new flavour-permutation-symmetric mixing observables. We give expressions for these plaquette invariants, both in terms of the mixing matrix elements alone, and in terms of manifestly Jarlskog-invariant functions of fermion mass matrices. While these quantities are unconstrained in the Standard Model, we point out that remarkably, in the case of leptonic mixing, the values of most of them are consistent with zero, corresponding to certain phenomenological symmetries. We give examples of their application to the flavour-symmetric description of both lepton and quark mixings, showing for the first time how to construct explicitly weak-basis invariant constraints on the mass matrices, for a number of phenomenologically valid mixing ansatze.