On universal quadratic identities for minors of quantum matrices

Abstract

We give a complete combinatorial characterization of homogeneous quadratic relations of "universal character" valid for minors of quantum matrices (more precisely, for minors in the quantized coordinate ring $O_q(M_{m,n}(K))$ of $m\times n$ matrices over a field $K$, where $q\in K^{\ast }$). This is obtained as a consequence of a study of quantized minors of matrices generated by paths in certain planar graphs, called SE-graphs, generalizing the ones associated with Cauchon diagrams. Our efficient method of verifying universal quadratic identities for minors of quantum matrices is illustrated with many appealing examples.

arXiv comment53 pages, 27 figures. This combines the (improved) original version and arXiv:1611.00302 [math.CO]. Submitted