# ALGEBRAIC COMBINATORICS

The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck polynomials
Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 187-208.

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 $\rho =\left(n,n-1,\cdots ,1\right)$ 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.199
Classification: 05E05
Keywords: Stembridge equality, Grothendieck polynomial, Young tableau, Hopf algebra.
Abney-McPeek, Fiona 1; An, Serena 2; Ng, Jakin S. 2

1 Harvard University Cambridge MA USA.
2 Massachusetts Institute of Technology Cambridge MA USA.
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Abney-McPeek, Fiona; An, Serena; Ng, Jakin S. The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck polynomials. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 187-208. doi : 10.5802/alco.199. https://alco.centre-mersenne.org/articles/10.5802/alco.199/

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