  The theory of Gabor frames of functions defined on finite abelian groups was
initially developed in order to better understand the properties of Gabor
frames of functions defined over the reals. However, during the last twenty
years the topic has acquired an interest of its own. One of the fundamental
questions asked in this finite setting is the existence of full spark Gabor
frames. The author proved the existence, as well as constructed such frames,
when the underlying group is finite cyclic. In this paper, we resolve the
non-cyclic case; in particular, we show that there can be no full spark Gabor
frames of windows defined on finite abelian non-cyclic groups. We also prove
that all eigenvectors of certain unitary matrices in the Clifford group in odd
dimensions generate spark deficient Gabor frames. Finally, similarities between
the uncertainty principles concerning the finite dimensional Fourier transform
and the short-time Fourier transform are discussed.

Rings and Algebras;Number Theory