We take a graph theoretic approach to the problem of finding generators for those prime ideals of $\mathcal{ O}_q(\mathcal{ M}_{ m,n}(\mathbb{ K}))$ which are invariant under the torus action ($\mathbb{ K}^*)^{ m+n}$. Launois \cite{ launois3} has shown that the generators consist of certain quantum minors of the matrix of canonical generators of $\mathcal{ O}_q(\mathcal{ M}_{ m,n}(\mathbb{ K}))$ and in \cite{ launois2} gives an algorithm to find them. In this paper we modify a classic result of Lindstr\"{ o}m \cite{ lind} and Gessel-Viennot~\cite{ gv} to show that a quantum minor is in the generating set for a particular ideal if and only if we can find a particular set of vertex-disjoint directed paths in an associated directed graph.
Categories: math.QA math.CO