On nonminimal |${\cal N}=4$|$N=4$ supermultiplets in 1D and their associated σ-models

We construct the nonminimal linear representations of the |${\cal N}=4$|$N=4$ extended supersymmetry in one-dimension. They act on eight bosonic and eight fermionic fields. Inequivalent representations are specified by the mass-dimension of the fields and the connectivity of the associated graphs. The oxidation to minimal |${\cal N}=5$|$N=5$ linear representations is given. Two types of |${\cal N}=4$|$N=4$ σ-models based on nonminimal representations are obtained: the resulting off-shell actions are either manifestly invariant or depend on a constrained prepotential. The connectivity properties of the graphs play a decisive role in discriminating inequivalent actions. These results find application in partial breaking of supersymmetric theories.