Top-degree components of Grothendieck and Lascoux polynomials

Abstract

The Castelnuovo-Mumford polynomial p Gw with w P Sn is the highest homogeneous component of the Grothendieck polynomial Gw. Pechenik, Speyer and Weigandt define a statistic rajcodep¨q on Sn that gives the leading monomial of p Gw. We introduce a statistic rajcodep¨q on any diagram D through a combinatorial construction "snow diagram" that augments and decorates D. When D is the Rothe diagram of a permutation w, rajcodepDq agrees with the aforementioned rajcodepwq. When D is the key diagram of a weak composition α, rajcodepDq yields the leading monomial of p Lα, the highest homogeneous component of the Lascoux polynomials Lα. We use p Lα to construct a basis of p Vn, the span of p Gw with w P Sn. Then we show p Vn gives a natural algebraic interpretation of a classical q-analogue of Bell numbers. Theorem 1.1 ([18]). Let w, u be permutations in S n . (A) The polynomial p G w has leading monomial x rajcodepwq . (B) We have p G w is a scalar multiple of p G u if and only if rajcodepwq " rajcodepuq. (C) If w is inverse fireworks (see §5), then x rajcodepwq has coefficient 1 in pG w . Moreover, there exists exactly one u 1 P S n that is inverse fireworks such that rajcodepuq " rajcodepu 1 q.Dreyer, Mészáros and St. Dizier [6] provide an alternative proof of (A) via the climbing chain model for Grothendieck polynomials introduced by Lenart, Robinson, and Sottile [15]. Hafner [9] provides an alternative proof of (A) for vexillary permutations via bumpless pipedreams.