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 $Y[\hat{s}_1,\hat{s}_2,\hat{s}_3]$ 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 $\Phi_{\mu^1[{\hat{s}_1}],\mu^2[{\hat{s}_2}], \mu^3[{\hat{s}_3}]}$ 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.