A graph theoretic method for determining generating sets of prime ideals in quantum matrices

We take a graph theoretic approach to the problem of finding generators for those prime ideals of Oq(Mm,n(K)) which are invariant under the torus action (K⁎)m+n. Launois (2004) [15], has shown that the generators consist of certain quantum minors of the matrix of canonical generators of Oq(Mm,n(K)) and in Launois (2004) [14], gives an algorithm to find them. In this paper we modify a classic result of Lindström (1973) [17], and Gessel and Viennot (1985) [6] 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.