Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 034, 57 pages      arXiv:1501.07337      https://doi.org/10.3842/SIGMA.2016.034

Notes on Schubert, Grothendieck and Key Polynomials

Anatol N. Kirillov abc
a) Research Institute of Mathematical Sciences (RIMS), Kyoto, Sakyo-ku 606-8502, Japan
b) The Kavli Institute for the Physics and Mathematics of the Universe (IPMU), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
c) Department of Mathematics, National Research University Higher School of Economics, 7 Vavilova Str., 117312, Moscow, Russia

Received March 26, 2015, in final form February 28, 2016; Published online March 29, 2016; Theorems 5.8 and 5.9 added and misprints corrected April 04, 2016

Abstract
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.

Key words: plactic monoid and reduced plactic algebras; nilCoxeter and idCoxeter algebras; Schubert, $\beta$-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 affine nilCoxeter algebras.

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