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

Abstract

We take a graph theoretic approach to the problem of finding generators for those prime ideals of O q ( M m , 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 O q ( M m , 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.