Flag-dipole spinors: On the dual structure derivation and $\mathcal{C}$, $\mathcal{P}$ and $\mathcal{T}$ symmetries

In this manuscript we report the flag-dipole spinors dual structure direct definition and analyze the properties behind the corresponding operator which generates such structure. This particular construction may be interesting for cosmological, phenomenological and mathematical physics applications. In addition, we analyse the behaviour of the flag-dipole spinors under action of discrete symmetries, facing an \emph{unconventional} property encoded on $(\mathcal{CPT})^2$.