Rectangle Diagrams for the Lusztig Cones of Quantized Enveloping Algebras of Type A

Let U be the quantum group associated to a Lie algebra g of type An. The negative part U− of U has a canonical basis B defined by Lusztig and Kashiwara, with favorable properties. We show how the spanning vectors of the cones defined by Lusztig (1993, Israel Math. Conf. Proc.7, 117–132), when regarded as monomials in Kashiwaraʹs root operators, can be described using a remarkable rectangle combinatorics. We use this to calculate the Lusztig parameters of the corresponding canonical basis elements, conjecturing that translates of these vectors span the simplicial regions of linearity of Lusztigʹs piecewise-linear function (1990, J. Amer. Math. Soc.3, 447–498, Sect. 2).