Combinatorial Rules for Three Bases of Polynomials

We present combinatorial rules (one theorem and two conjectures) concerning three bases of Z[x1, x2, ....]. First, we prove a ”splitting” rule for the basis of Key polynomials [Demazure ’74], thereby establishing a new positivity theorem about them. Second, we introduce an extension of [Kohnert ’90]’s ”moves” to conjecture the first combinatorial rule for a certain deformation [Lascoux ’01] of the Key polynomials. Third, we use the same extension to conjecture a new rule for the Grothendieck polynomials [LascouxSchützenberger ’82]. In memory of Alain Lascoux, who inspired this paper one night in Osaka