On Lagrangian formulations for (ir)reducible mixed-antisymmetric higher integer spin fields in Minkowski spaces

We extend the results of Lagrangian formulations study to construct gauge-invariant Lagrangians for (ir)reducible integer higher-spin massless and massive representations of the Poincare group with a Young tableau $Y[{\widehat{s}}_1,{\widehat{s}}_2,{\widehat{s}}_3]$ in $d$-dimensional flat space-time (as the probable candidates to describe the Dark Matter problem beyond the SM). These particles are described within a metric-like formulation by tensor fields with 3 groups of antisymmetric Lorentz indices ${\mathrm{\Phi }}_{{\mu }^1[{\widehat{s}}_1],{\mu }^2[{\widehat{s}}_2],{\mu }^3[{\widehat{s}}_3]}$ on a basis of the BRST method with complete, $Q$, and incomplete, $Q_c$, BRST operators. We found unconstrained (with $Q$) and constrained (with $Q_c$ and off-shell BRST invariant holonomic constraints) gauge Lagrangian formulations with different configuration spaces and reducibility stages. The deformation procedure to construct interacting gauge model with mixed-antisymmetric fields is proposed.