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.

Hecke-Grothendieck 多项式由 Kirillov 引入，作为 Schubert 多项式、对偶α- Grothendieck 多项式、Di Francesco-Zinn-Justin 多项式等的通用推广。 然后 Kirillov 推测每个广义 Hecke-Grothendieck 多项式的系数是某些特定的非负组合参数。这里我们证明了基里洛夫猜想的弱版本，即在一定条件下，每个 Hecke-Grothendieck 多项式只有非负整数系数。特别是，这个弱版本的基里洛夫猜想的证明可以作为所有舒伯特多项式、对偶α -Grothendieck 多项式和 Di Francesco-Zinn-Justin 多项式只有非负系数的事实的统一证明。