Quantum matrices by paths
Karel Casteels · Algebra & Number Theory · 2014
We study, from a combinatorial viewpoint, the quantized coordinate ring of m n matrices ᏻ q .ᏹm;n //ދ. over an infinite field ދ (often simply called quantum matrices).The first part of this paper shows that ᏻ q .ᏹm;n ,//ދ.which is traditionally defined by generators and relations, can be seen as a subalgebra of a quantum torus by using paths in a certain directed graph.Roughly speaking, we view each generator of ᏻ q .ᏹm;n //ދ. as a sum over paths in the graph, each path being assigned an element of the quantum torus.The ᏻ q .ᏹm;n //ދ.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 applies the above to the theory of torus-invariant prime ideals of ᏻ q .ᏹm;n .//ދ.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 when char./ދ D 0 and with q transcendental over .ޑOur strategy is to prove the stronger result that the quantum minors in a given torusinvariant ideal form a Gröbner basis.