Proof of a conjectured Möbius inversion formula for Grothendieck polynomials (Q6619823)

The flag variety \(\mathcal{F}\ell_n\) is the module space of complete flags \(V_0 \subset V_1 \subset \ldots \subset V_n=\mathbb{C}^n\) of nested vector subspaces of \(\mathbb{C}^n\), where \(\dim V_i = i\) for each \(i\). The flag variety is stratified by its Schubert varieties \(X_w\) for \(w\in S_n\). Taking Poincaré duals of these subvarieties yields a distinguished Schubert basis of the integral cohomology ring \(H^*(\mathcal{F}\ell_n)\). Taking classes of their structure sheaves gives a distinguished Schubert basis of the \(K\)-theory ring \(K^0(\mathcal{F}\ell_n)\).\N\NSchubert polynomials \(\mathfrak{S}_w\) are polynomial representatives for cohomology classes of Schubert varieties in a complete flag variety, while Grothendieck polynomials \(\mathfrak{G}_w\) are analogous representatives for the \(K\)-theory classes of the structure sheaves of Schubert varieties. In the special case that \(\mathfrak{S}_w\) is a multiplicity-free sum of monomials, \(\mathfrak{G}_w\) can be easily computed from \(\mathfrak{S}_w\) via Möbius inversion on a certain poset. O. Pechnik and M. Satriano prove this conjecture by realizing monomials as Chow classes on a product of projective spaces and invoke a result of M. Brion on flat degenerations of such classes.