  In this paper we introduce several computational techniques for the study of
moduli spaces of McKay quiver representations, making use of Groebner bases and
toric geometry. For a finite abelian group G in GL(n,k), let Y_\theta be the
coherent component of the moduli space of \theta-stable representations of the
McKay quiver. Our two main results are as follows: we provide a simple
description of the quiver representations corresponding to the torus orbits of
Y_\theta, and, in the case where Y_\theta equals Nakamura's G-Hilbert scheme,
we present explicit equations for a cover by local coordinate charts. The
latter theorem corrects the first result from [Nakamura]. The techniques
introduced here allow experimentation in this subject and give concrete
algorithmic tools to tackle further open questions. To illustrate this point,
we present an example of a nonnormal G-Hilbert scheme, thereby answering a
question raised by Nakamura.
