Constrained BRST-BFV and BRST-BV Lagrangians for half-integer HS fields on $R^{1,d-1}$

Gauge invariant Lagrangian descriptions of irreducible and reducible half-integer higher-spin mixed-symmetric massless and massive representations of the Poincare group with off-shell algebraic constraints are constructed within a metric-like formulation in a $d$-dimensional flat space-time on the basis of a suggested constrained BRST approach. A Lorentz-invariant resolution of the BRST complex within the constrained BRST formulations produces a gauge-invariant Fang-Fronsdal Lagrangian entirely in terms of the initial triple gamma-traceless spin-tensor field $\Psi_{(\mu)_{n}}$ with gamma-traceless gauge parameter. The triplet and quartet formulations are derived. The minimal (un)constrained BRST--BV actions for above formulations are obtained, from proposed constrained BRST--BV approach to be by appropriate tools to construct interacting constrained Lagrangians.