  We consider the double-row (open) transfer matrix constructed from generic
tensor-type representations of various types of Hecke algebras. For different
choices of boundary conditions for the relevant integrable lattice model we
express the double-row transfer matrix solely in terms of generators of the
corresponding Hecke algebra (tensor-type realizations). We then expand the open
transfer matrix and extract the associated Murphy elements from the first/last
terms of the expansion. Suitable combinations of the Murphy elements as has
been shown commute with the corresponding Hecke algebra.
