  Flat energy bands of model lattice Hamiltonians provide a key ingredient in
designing dispersionless wave excitations and have become a versatile platform
to study various aspects of interacting many-body systems. Their essential
merit lies in hosting compactly localized eigenstates which originate from
destructive interference induced by the lattice geometry, in turn often based
on symmetry principles. We here show that flat bands can be generated from a
hidden symmetry of the lattice unit cell, revealed as a a permutation symmetry
upon reduction of the cell over two sites governed by an effective dimer
Hamiltonian. This so-called latent symmetry is intimately connected to a
symmetry between possible walks of a particle along the cell sites, starting
and ending on each of the effective dimer sites. The summed amplitudes of any
eigenstate with odd parity on the effective dimer sites vanish on special site
subsets called walk multiplets. We exploit this to construct flat bands by
using a latently symmetric unit cell coupled into a lattice via walk multiplet
interconnections. We demonstrate that the resulting flat bands are tunable by
different parametrizations of the lattice Hamiltonian matrix elements which
preserve the latent symmetry. The developed framework may offer fruitful
perspectives to analyze and design flat band structures.
