Notes on Schubert, Grothendieck and Key Polynomials

[摘要] We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.

[关键词] plactic monoid and reduced plactic algebras;nilCoxeter and idCoxeter algebras;Schubert;β-Grothendieck;key and (double) key-Grothendieck;and Di Francesco– Zinn-Justin polynomials;Cauchy’s type kernels and symmetric;totally symmetric plane partitions;and alternating sign matrices;noncrossing Dyck paths and (rectangular) Schubert polynomials;multi-parameter deformations of Genocchi numbers of the first and the second types;Gandhi–Dumont polynomials and (staircase) Schubert polynomials;double af fine nilCoxeter algebras