Linear Dependency of Translations and Square Integrable Representations

Let $G$ be a locally compact group. We examine the problem of determining when nonzero functions in $L^2(G)$ have linearly independent translations. In particular, we establish some results for the case when $G$ has an irreducible, square integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group and the Weyl-Heisenberg group. We also investigate the case when $G$ has an abelian, closed subgroup of finite index.