  We discuss how the shift operator and the Hamiltonian enter the hierarchy of
Baxter Q-operators in the example of gl(n) homogeneous spin-chains. Building on
the construction that was recently carried out by the authors and their
collaborators, we find that a reduced set of Q-operators can be used to obtain
local charges. The mechanism relies on projection properties of the
corresponding R-operators on a highest/lowest weight state of the quantum
space. It is intimately related to the ordering of the oscillators in the
auxiliary space. Furthermore, we introduce a diagrammatic language that makes
these properties manifest and the results transparent. Our approach circumvents
the paradigm of constructing the transfer matrix with equal representations in
quantum and auxiliary space and underlines the strength of the Q-operator
construction.
