ALGEBRAIC COMBINATORICS

no. 2
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, Volume 5 (2022) no. 2, pp. 187-208.

The Schur polynomials s λ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For ρ=(n,n-1,⋯,1) a staircase shape and μ⊆ρ a subpartition, the Stembridge equality states that s ρ/μ =s ρ/μ T . This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials G λ , and the dual stable Grothendieck polynomials g λ , 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 ρ/μ =G ρ/μ T and g ρ/μ =g ρ/μ T , the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials.

Received:
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.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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