Quantum Matrices by Paths

TL;DR

This paper introduces a combinatorial path-based approach to quantum matrices, proving that all torus-invariant prime ideals are generated by quantum minors when q is not a root of unity.

Contribution

It provides a new path-based perspective on quantum matrices and proves a conjecture that all torus-invariant prime ideals are generated by quantum minors for non-root of unity q.

Findings

Quantum matrices can be represented via paths in a directed graph.

All torus-invariant prime ideals are generated by quantum minors when q is not a root of unity.

The path viewpoint relates to Cauchon's deleting-derivations algorithm.

Abstract

We study, from a combinatorial viewpoint, the quantized coordinate ring of mxn matrices over an infinite field K (also called quantum matrices) and its torus-invariant prime ideals. The first part of this paper shows that this algebra, traditionally defined by generators and relations, can be seen as subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of quantum matrices as a sum over paths in the graph, each path being assigned an element of the quantum torus. The quantum matrices relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchon's deleting-derivations algorithm. The second part of this paper is to apply the paths viewpoint to the theory of torus-invariant prime ideals of quantum matrices. We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter q is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only for char(K)=0 and q transcendental over Q. Our strategy is to show that the quantum minors in a given torus-invariant ideal form a Grobner basis.