Quasi-commuting families of quantum minors

In [B. Leclerc, A. Zelevinsky, Quasicommuting families of quantum Plücker coordinates, in: Kirillovʹs 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 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 to quasi-commute. In addition, we study the combinatorics of maximal (by inclusion) families of pairwise quasi-commuting quantum minors and pose relevant conjectures.