There is a two-part programme to prove the existence of degenerations of Schubert varieties of \(\mathrm{SL}(n)\) into toric varieties, see for instance [N. Gonciulea, and V. Lakshmibai, Transform. Groups 1, 215–248 (1996; Zbl 0909.14028)]. Here the authors present results concerning the first of the two parts. They prove that for a certain class of elements \(w\) of the Weyl group of \(sl_{n+1}\), there exists a lattice polytope \(\Delta_i^w \subset {\mathbb{R}}^{l(w)}\), such that for every dominant weight \(\lambda = \sum_{i = 1}^na_iw_i\) where the \(w_i\) are fundamental weights the number of lattice points in the Minkowski sum \(\Delta_\lambda^w = \sum_{i=1}^na_i\Delta_i^w\) is equal to the dimension of the Demazure module \(E_w(\lambda)\) (for its definition see [M. Demazure, Bull. Sci. Math., II. Ser. 98, 163–172 (1975; Zbl 0365.17005)]. They also present an explicit formula which gives the character of \(E_w(\lambda)\).
For \(w = w_0\) the longest element of the Weyl group other polytopes satisfying the same condition have been constructed by A. D. Berenstein and A. Zelevinsky [J. Geom. Phys. 5, 453–472 (1988; Zbl 0712.17006)] and P. Littelmann [Transform. Groups 3, 145–179 (1998; Zbl 0908.17010)]. The authors of the present paper believe that their construction can be generalized to every simple algebraic group.