Combinatorial Formulae for Grothendieck-demazure and Grothendieck Polynomials

∂if = f− sif xi − xi+1 where si acts on f by transposing xi and xi+1 and let π̃i = ∂i(xi(1− xi+1)f) Then the Grothendieck-Demazure polynomial κα, which is attributed to A. Lascoux and M. P. Schützenberger, is defined as κα = x α1 1 x α2 2 x α3 3 ... if α1 ≥ α2 ≥ α3 ≥ ..., i.e. α is non-increasing, and κα = π̃iκαsi if αi < αi+1, where si acts on α by transposing the indices. Example 2.1. Let α = (0, 2, 1). Then