  We propose a new tool, which we call $M$-decompositions, for devising
superconvergent hybridizable discontinuous Galerkin (HDG) methods and
hybridized-mixed methods for linear elasticity with strongly symmetric
approximate stresses on unstructured polygonal/polyhedral meshes.
  We show that for an HDG method, when its local approximation space admits an
$M$-decomposition, optimal convergence of the approximate stress and
superconvergence of an element-by-element postprocessing of the displacement
field are obtained. The resulting methods are locking-free.
  Moreover, we explicitly construct approximation spaces that admit
$M$-decompositions on general polygonal elements. We display numerical results
on triangular meshes validating our theoretical findings.
