General Lagrangian formulations for mixed-antisymmetric tensor fields on flat backgrounds

Lagrangian formulations for (ir)reducible integer higher-spin massless and massive Poincare group representations subject to Young tableau with $k$ columns $Y[{\widehat{s}}_1,{\widehat{s}}_2,\mathrm{.}\mathrm{.}\mathrm{.},{\widehat{s}}_k]$ in $d$-dimensional Minkowski space-time are firstly presented. The particles are described in a metric-like formulation by tensor fields with $k$ groups of antisymmetric Lorentz indices ${\mathrm{\Phi }}_{{\mu }^1[{\widehat{s}}_1],{\mu }^2[{\widehat{s}}_2],\mathrm{.}\mathrm{.}\mathrm{.},{\mu }^k[{\widehat{s}}_k]}$ by means of the BRST procedure with complete, $Q$, and incomplete, $Q_c$, BRST operators. Starting from a description of bosonic mixed-antisymmetric higher-spin fields in terms of an auxiliary Fock space associated with a special Poincare module, we realize a conversion of the initial operator constraint system into a system of first-class operator constraints. To this aim, we find, in first time, by means of Verma module the auxiliary representations of the constraint subalgebra, to be isomorphic due to Howe duality to $so(k,k)$ algebra, and containing the subsystem of second-class operators in terms of new oscillator variables forming the Fock module. An unconstrained (with $Q$) and constrained (with $Q_c$ and BRST invariant algebraic constraints) gauge Lagrangian formulations with equivalent dynamics, but different configuration spaces are found. Concept of consistent interactions are suggested.