Bijection with the Newton–Okounkov polytope

Let $y^t cin coperatorname{Conf}^{cdagger,+}_n V'(cmathbb{Z}^t)(a_1,\cdots,a_n)$, and define

$$b_i=csum_{j<i}a_j.$$

Act on $y^t$ by the element

$$cfrac{1}{k}(b_1,\cdots,b_n)cin T^n(cmathbb{Z}^t).$$

The proposed bijection. The resulting point is the corresponding point in $NO_G^r$.