A weak version of Kirillov's conjecture on Hecke–Grothendieck polynomials

Abstract

Hecke-Grothendieck polynomials were introduced by Kirillov as a common generalization of Schubert polynomials, dual α-Grothendieck polynomials, Di Francesco–Zinn–Justin polynomials, etc. Then Kirillov conjectured that the coefficients of every generalized Hecke-Grothendieck polynomial are nonnegative combinations of certain parameters. Here we prove a weak version of Kirillov's conjecture, that is, under certain conditions, every Hecke-Grothendieck polynomial has only nonnegative integer coefficients. In particular, the proof of this weak version of Kirillov's conjecture serves as a unified proof for the fact that all the Schubert polynomials, dual α-Grothendieck polynomials, and Di Francesco–Zinn–Justin polynomials have only nonnegative coefficients.