  We begin by deriving an action of the 0-Hecke algebra on standard reverse
composition tableaux and use it to discover 0-Hecke modules whose
quasisymmetric characteristics are the natural refinements of Schur functions
known as quasisymmetric Schur functions. Furthermore, we classify
combinatorially which of these 0-Hecke modules are indecomposable.
  From here, we establish that the natural equivalence relation arising from
our 0-Hecke action has equivalence classes that are isomorphic to subintervals
of the weak Bruhat order on the symmetric group. Focussing on the equivalence
classes containing a canonical tableau we discover a new basis for the Hopf
algebra of quasisymmetric functions, and use the cardinality of these
equivalence classes to establish new enumerative results on truncated shifted
reverse tableau studied by Panova and Adin-King-Roichman.
  Generalizing our 0-Hecke action to one on skew standard reverse composition
tableaux, we derive 0-Hecke modules whose quasisymmetric characteristics are
the skew quasisymmetric Schur functions of Bessenrodt et al. This enables us to
prove a restriction rule that reflects the coproduct formula for quasisymmetric
Schur functions, which in turn yields a quasisymmetric branching rule analogous
to the classical branching rule for Schur functions.
