  Subword complexes are simplicial complexes introduced by Knutson and Miller
to illustrate the combinatorics of Schubert polynomials and determinantal
ideals. They proved that any subword complex is homeomorphic to a ball or a
sphere and asked about their geometric realizations. We show that a family of
subword complexes can be realized geometrically via regular triangulations of
root polytopes. This implies that a family of $\beta$-Grothendieck polynomials
are special cases of reduced forms in the subdivision algebra of root
polytopes. We can also write the volume and Ehrhart series of root polytopes in
terms of $\beta$-Grothendieck polynomials.
