A Graph Theoretic Method for Determining Generating Sets of Prime Ideals in Quantum Matrices

TL;DR

This paper introduces a graph theoretic approach to identify generators of certain prime ideals in quantum matrices, linking algebraic generators to vertex-disjoint paths in an associated graph.

Contribution

It adapts classical combinatorial results to quantum algebra, providing a new method to determine generators of invariant prime ideals in quantum matrix algebras.

Findings

Quantum minors are in the generating set if specific vertex-disjoint paths exist.

The method simplifies finding generators by graph-theoretic criteria.

Provides an algorithmic approach based on graph paths.

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 ($\mathbb{K}^*)^{m+n}$. 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.