Density matrices in integrable face models

Using the properties of local Boltzmann weights integrable interaction-round-a-face (IRF or face) models we express operators in terms generalized transfer matrices. This allows for derivation discrete functional equations reduced density matrices inhomogeneous generalizations these models. We apply to study IRF various solid-on-solid type and quantum chains non-Abelian $\bm{su(2)_3}$ Fibonacci anyons. Similar as six vertex model find that a sequence consecutive sites can be 'factorized', i.e.\ expressed nearest-neighbour correlators with coefficients which are independent parameters. Explicit expressions provided correlation functions on up three neighbouring sites.