  Nonlinear lattice models can support "discrete breather" excitations that
stay localized in space for all time. By contrast, the localized Wannier states
of linear lattice models are dynamically unstable. Nevertheless, symmetric and
exponentially localized Wannier states are a central tool in the classification
of band structures with crystalline symmetries. Moreover, the quantized
transport observed in nonlinear Thouless pumps relies on the fact that -- at
least in a specific model -- discrete breathers recover Wannier states in the
limit of vanishing nonlinearity. Motivated by these observations, we
investigate the correspondence between nonlinear breathers and linear Wannier
states for a family of discrete nonlinear Schr\"odinger equations with
crystalline symmetries. We develop a formalism to analytically predict the
breathers' spectrum, center of mass and symmetry representations, and apply
this to nonlinear generalizations of the Su-Schrieffer-Heeger chain and the
breathing kagome lattice.
