The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck polynomials

Abney-McPeek, Fiona¹ ; An, Serena² ; Ng, Jakin S.²

¹ Harvard University Cambridge MA USA.
² Massachusetts Institute of Technology Cambridge MA USA.

Algebraic Combinatorics, Tome 5 (2022) no. 2, pp. 187-208.

Résumé

The Schur polynomials $s_{\lambda }$ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For $\rho =(n,n-1,\cdots ,1)$ a staircase shape and $\mu \subseteq \rho$ a subpartition, the Stembridge equality states that $s_{\rho /\mu }=s_{\rho /{\mu }^T}$. This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials $G_{\lambda }$, and the dual stable Grothendieck polynomials $g_{\lambda }$, developed by Buch, Lam, and Pylyavskyy, are variants of the Schur polynomials and describe the $K$-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood–Richardson rule, we prove that $G_{\rho /\mu }=G_{\rho /{\mu }^T}$ and $g_{\rho /\mu }=g_{\rho /{\mu }^T}$, the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials.

Détail

Reçu le : 2021-03-23
Révisé le : 2021-10-02
Accepté le : 2021-10-06
Publié le : 2022-05-05

DOI : 10.5802/alco.199

Classification : 05E05
Keywords: Stembridge equality, Grothendieck polynomial, Young tableau, Hopf algebra.

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Bibliographie

[1] Alwaise, Ethan; Chen, Shuli; Clifton, Alexander; Patrias, Rebecca; Prasad, Rohil; Shinners, Madeline; Zheng, Albert Coincidences among skew stable and dual stable Grothendieck polynomials, Involve, Volume 11 (2018) no. 1, pp. 143-167 | Zbl | MR | DOI

[2] Buch, Anders S. A Littlewood–Richardson rule for the $K$-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78 | Zbl | MR | DOI

[3] Fomin, Sergey; Kirillov, Anatol N. Grothendieck polynomials and the Yang–Baxter equation, Formal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, DIMACS, Piscataway, NJ, sd, pp. 183-189 | MR

[4] Galashin, Pavel A Littlewood–Richardson rule for dual stable Grothendieck polynomials, J. Combin. Theory Ser. A, Volume 151 (2017), pp. 23-35 | Zbl | MR | DOI

[5] Grinberg, Darij; Reiner, Victor Hopf Algebras in Combinatorics (2020) (https://arxiv.org/abs/1409.8356)

[6] Lam, Thomas; Pylyavskyy, Pavlo Combinatorial Hopf algebras and K-homology of Grassmanians, Int. Math. Res. Not. (2007) no. 24, rnm125, 48 pages | DOI

[7] Reiner, Victor; Shaw, Kristin M.; van Willigenburg, Stephanie Coincidences among skew Schur functions, Adv. Math., Volume 216 (2007) no. 1, pp. 118-152 | Zbl | MR | DOI

[8] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, 2012, xiv+626 pages (second edition) | Zbl | MR

[9] Yeliussizov, Damir Duality and deformations of stable Grothendieck polynomials, J. Algebraic Combin., Volume 45 (2017) no. 1, pp. 295-344 | Zbl | MR | DOI