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 ${\mathcal{O}}_q({\mathcal{M}}_{m,n}(\mathbb{K}))$ which are invariant under the torus action (${\mathbb{K}}^{\ast }{)}^{m+n}$. 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 Lindström \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.

29 pages, 9 figures

References in corpus (4)

• Total positivity, Grassmannians, and networks
• Combinatorial formulas for Le-coordinates in a totally nonnegative Grassmannian
• Totally nonnegative cells and matrix Poisson varieties
• An automaton-theoretic approach to the representation theory of quantum algebras