  The fusion procedure provides a way to construct new solutions to the
Yang-Baxter equation. In the case of the symmetric group the fusion procedure
has been used to construct diagonal matrix elements using a decomposition of
the Young diagram into its rows or columns. We present a new construction which
decomposes the diagram into hooks, the great advantage of this is that it
minimises the number of auxiliary parameters needed in the procedure. We go on
to use the hook fusion procedure to find diagonal matrix elements
computationally and calculate supporting evidence to a previous conjecture.
  We are motivated by the construction of certain elements that allow us to
generate representations of the symmetric group and single out particular
irreducible components. In this way we may construct higher representations of
the symmetric group from elementary ones. We go some way to generalising the
hook fusion procedure by considering other decompositions of Young diagrams,
specifically into ribbons. Finally, we adapt our construction to the quantum
deformation of the symmetric group algebra known as the Hecke algebra.
