We study, from a combinatorial viewpoint, the
quantized coordinate ring ofmatrices
over an
infinite field
(often simply called
quantum matrices).The first part of this paper shows that
, which is
traditionally defined by generators and relations, can be seen as a subalgebra of a quantum
torus by using paths in a certain directed graph. Roughly speaking, we view each generator
of
as a
sum over paths in the graph, each path being assigned an element of the quantum torus.
The
relations then arise naturally by considering intersecting paths. This viewpoint is
closely related to Cauchon’s deleting derivations algorithm.
The second part of this paper applies the above to the theory of torus-invariant prime
ideals of
. We
prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum
parameter
is a non-root of unity, have generating sets consisting of quantum
minors. Previously, this result was known to hold only when
and with
transcendental
over
.
Our strategy is to prove the stronger result that the quantum minors in a given
torus-invariant ideal form a Gröbner basis.
