Toward Berenstein-Zelevinsky data in affine type $A$, part II: Explicit   description
  In the present paper, we give an explicit description of the affine analogs of Berenstein-Zelevinsky data constructed in our previous paper: Toward Berenstein-Zelevinsky data in affine type $A$, I: Construction of affine analogs (arXiv:1009.4526), in terms of certain collections of nonnegative integers, which we call Lusztig data of type $A_{l-1}^{(1)}$. 
Satoshi Naito, Daisuke Sagaki, and Yoshihisa Saito
math.QA math.RT