Quasi-commuting families of quantum minors

In [B. Leclerc, A. Zelevinsky, Quasicommuting families of quantum Plücker coordinates, in: Kirillov's Seminar on Representation Theory, in: Amer. Math. Soc. Transl. (2), vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 85-108], a combinatorial criterion for quasi-commutativity is established for pairs of quantum Plücker coordinates in the quantized coordinate algebra Cq[F] of the flag variety of type A. This paper attempts to generalize these results by producing necessary and sufficient conditions for pairs of quantum minors in the quantized coordinate algebra Cq[Matk×m] to quasi-commute. In addition, we study the combinatorics of maximal (by inclusion) families of pairwise quasi-commuting quantum minors and pose relevant conjectures.